Complexification And The Wave Function Of The Universe

[Aethelwulf]

Time Evolution, the Wave Function of the Universe and Complexification

 

 

 

In a previous essay http://scienceforums.com/topic/26897-true-time-evolution-in-gr-and-the-wheeler-de-witt-equation/, I loosely spoke about general concepts of time, how we often view time and what science has to say about time. We uncovered several possible identities for time, those being fundamental, geometrical, local, global, psychological and perhaps an induced time.

 

It was clearly stated in the previous essay, that it was taken by the author (me), that time is not fundamental. For it to be fundamental, it needs to have an appearance when the universe came into existence. The deep problem with this is of course heavily rooted in the relativistic disciplines of the 20th and 21st century, that being that time itself is generally considered a dimension of space. Taking this relativistic principle seriously, then time itself fails to be fundamental because of our universe is by current theory, generally considered to appear from a single point with no degrees of freedom.

 

I showed, that normally in GR (General Relativity), this tiny (dimensionless) region of spacetime is where an infinite curvature can be present. This statement is itself contradictory, you can not deal with a pointlike system and then say that system possesses an unrealistic and unphysical singularity. This is stated as a serious problem in this essay for the unification of physics and relativity at the very small scales and the beginning of the universe itself.

 

There are however some models existing now which can deal with this singularity and smooth out these infinities. One such example is the extension of GR called the Cartan-Einstein model, which represents GR as the full Poincare group allowing new gravitational influences in the theory, such as torsion itself - torsion was neglected originally by Einstein because he could not find an electromagnetic theory for it, but today there have been major achievements towards this area of research. It also removes any infinities at the Big Bang and a collaborating paper can be found at the very last post of my last essay - 'how a universe can come from nothing and still not have a singularity.'

 

 

In my previous essay, the problem of time, is that on the global sense, the universe itself is governed by timeless laws. The only way to interpret that from General Relativity, is by stating that the wave function of the universe (and the dynamics it governs) when quantized yield situations where time will show itself not to be a fundamental aspect of the universe. An interesting problem (from which I have been heavily influenced recently by Julian Barbour) is the problem of complexification for this wave function of the universe - it simply is not complex.

 

The rule which governs this timeless law is stated through the Wheeler de-Witt Equation WDW

 

[math]H|\psi > = 0[/math]

 

On the right the time derivative vanishes, what we are left with is a type of time-independant solution, similar to a Schrodinger time independent case of his wave equation.

 

In the ideal world, we would not only have a time derivative, we would also have complexification acting as a coefficient on [math](\frac{\partial}{\partial t})[/math]. When we talk about complexifying fields, it can mean quite a few things. Let's begin with an example which can be given concerning our formulation.

 

We would have a Hamiltonian which contains a constraint:

 

[math]\pi_t + H = 0[/math]

 

[math]\pi_t[/math] is the momentum conjugate to time (it's not a true conjugate) [1], it is like it. To get a quantum theory from this classical constraint into a wave equation, [math]\pi_t[/math] would become [math]i \frac{\partial}{\partial t}[/math]. From there you would end up with the Schrodinger equation. From here, Barbour has made a very important point in our usual understanding of quantum mechanics... that being ''quantum mechanics is inherently complex.''

 

If one however quantizes the field equations describing a closed universe, you end up with a time-independent solution which is not even complex, the Wheeler de-Witt equation. Basically, you apply Dirac's canonical quantization methods to GR to reveal this non-complex probability field.

 

An equation of this description is always obtained when you treat a closed universe without a reference to an external time. The interpretation of this, seems to state that we live in a timeless, or some prefer a ''static'' universe which depends on no time but only three dimensions of freedom we know intuitively as the three dimensions of space.

 

But there is a further problem with this - the Wheeler de-Witt equation must be further incomplete when you consider the beginning of the universe, since there was not even spatial dimensions to speak about. In Barbour's research, he concluded that the complexity of quantum theory is inseparably associated with time. Some people have offered solutions to re-establishing a complex Wave function of the universe. One such attempt might be reconfiguring the wave function under the Transactional Interpretation, which splits the wave function into a symmetric echo-offer wave distribution which is spoke about in my previous essay. However, this doesn't in essence ''fix'' the problem of how time is equally non-fundamental as space is at the very beginning set-up conditions of our universe. Assuming the Big Bang is correct, then there is a problem with the Wheeler de-Witt equation as well in the treatment of space itself.

 

The basic equation of quantum theory describing gravity may be real and not complex - however, Barbour mentions that there is a serious study which would suggest that gravity physics will require a complex fundamental equation. I however, find the WDW equation to be somewhat of an interesting hint on the breakdown of General Relativity for the fundamental roles of the universe. It is also a breakdown itself by still permitting three dimensions of space, when neither space itself is fundamental. Barbour does state that one can fix the equations not to treat the universe with an external time, but one with an internal time. Later in his works, he shows that time does not even need to exist globally, but is measured rather by the individual motions of all objects contained within the universe. Time is however, I will add to this, measured by clocks, so time certainly appears to emergent, or induced when geometry begins to appear in the universe when it has sufficiently cooled down. If one was remove the three dimensional case out of the WDW equation, what we would have then is something similar to the universe as it was when it first appeared - there was no space, no time and with no global description of time, the conservation of energy can be violated as the universe evolves.

 

Whowever can solve this problem concerning the Wheeler de-Witt equation, would in my books deserve a nobel prize.

 

 

[1] - The Hamiltonian is a generator of a systems evolution. It can also be seen as a conjugate momentum to time, meaning that time is thought of a generalized coordinate, the momentum corresponding to that coordinate will be the Hamiltonian. This is akin to the momentum corresponding to a Cartesian coordinate.

Edited November 11, 2012 by Aethelwulf

[Aethelwulf]

All Canonical theories of gravity which have been quantized have to deal with the time problem. I have explained that, if the WDW equation is an equation describing the whole universe, then it is at odd's with how our current theory treats the initial conditions of our universe. Our universe is not fundamentally-concerned with geometry, then neither should a quantized version of the relativistic laws. Time disappears in the Canonical theories of gravity, but the space dimensions are normally retained. That simply cannot have real physical applications when you wind the ''imaginary'' clock of the universe back to the very first instant.

 

Just revisiting very quickly, one problem concerning this area of research is whether the WDW equation should be complixified, and if so, what is the best approach? Should we reconfigure time into the equations? After all, would not such a theory still break down when geometry ceases to exist in a closed universe?

 

Concerning the first part, most physicists would tend to think the equation should be complex. Since we have established from the work of Barbour, that the complex nature is inseparable to time itself, we should note that almost all of the fundamental laws of nature we can write about are in fact time-symmetric. Such as radiation laws, electromagnetism has retarded and advanced solutions, thermodynamics is also a good example of time-symmetric laws, thus the theory of gravity is often thought to have a symmetric case involving time as well.

 

 

What we have is a real and present problem with the dogma of time - it is generally considered that time itself is needed in a quantum mechanical or field theory, but this is not the case. Barbour himself was one of the main proponents to show the world how, our world is governed by change, not by time. He was led on this crusade by Mach, which he himself has often quoted

 

“It is utterly beyond our power to measure the changes of things by time. Quite the contrary, time is an abstraction at which we arrive through the changes of things.”

 

Anyone who is a little versed in the history of relativity are also aware of what a major influence Mach had been to Einstein's train of thought and perhaps, this is one reason why his relativistic theory turned out to be governed by timeless equations. Einstein's GR seriously puts into doubt the idea of time dimension, when you realize that time evolution arises as a symmetry of the theory, it is not like a true time evolution at all.

 

However, the Machian view of time is slightly different to the Einsteinian time found in the special theory. Machian time arises as points which are relative to each other in space, in Einstein's view, you move through time as ''splits'' as you move through space. However, the Machian view does cast doubt in Einstein's relativity and is also at the core of what Barbour believes; that being there is no time, only motion of objects relative to each other. Dirac once said

 

''I am inclined to believe that four-dimensional symmetry is not a fundamental property of the physical world.''

 

I agree with Dirac. Space nor time as symmetries, or space or time alone are fundamental properties of the universe itself when you probe the universe to it's very origins.

[Aethelwulf]

What models can I use then which could help describe this initial condition of the universe? The model I have proposed requires that:

 

 

1) Space and time are not fundamental objects

 

2) That matter appears in the low energy phase of the universe when geometry appears (Geometrogenesis)

 

3) That the universe does not conserve or even have a well defined energy. Indeed, I have pointed out that the universe may not even have an energy.

 

 

The third problem is a measurement problem, the universe simply cannot have an energy unless someone has measured it.

 

The second problem is rooted in the idea that geometry and matter appeared when the universe had sufficiently cooled down.

 

And the first problem has to do with our current theory on Big Bang - that it began as a point with no dimensions at all.

 

Triangulation and Graphical Tensor Notation

 

I am going to explain the best direction I think to prep a theory for all three conditions. The first two are really not too problematic, the first one is a condition we have to set out theory in about the initial state of the universe. Knowing off-hand that space and time are not truly fundamental is a tricky thing to wrap your head roung mind you. The second statement is about what we know about the physical laws of the universe, like the basics of nucleosynthesis and when matter and geometry appears.

 

The third problem is the greatest problem, because we need a model which can sufficiently state that perhaps energy was zero at the initial condition of the universe - only as the universe expanded, energy began to appear.

 

So, to quickly get through this, what is the best model to suit statement 3?

 

The answer is spin dynamics and triangulation. In spin dynamics, you can triangulate a kind of geometry on your fibre bundle (which looks like tiny segments on your manifold). Each point of the triangle corresponds to the location of a single particle. Being a completely relativistic thing, energy then is measured by the locations of these objects. One point alone will not have an energy, only another object for instance relative to another object can graph out an energy.

 

[math]A(G)[/math] are adjacent vertices and [math]E(G)[/math] are the sets of edges found in the configuration space. To find the energy in our graph, you use the equation

 

[math]E(G) = <\psi_G|H|\psi_G>[/math]

 

Basically, energy arises as a relativistic principle in this specific configuration space. So what happens when you reach zero size at the initial start-up condition of the universe?

 

All the lengths of your causal triangle go to zero - if there are no lengths between objects, then you cannot have a notion or quantity of energy in a relativistic sense. Note, that this idea ties in very well with our current idea's on how the universe expands - that is, in the beginning, space appeared between the objects, it is space which expands. The geometry made sense when you have this geometry appear in this combinatorial fashion (the triangulation of matter).

 

Therefore, energy was necessarily zero, space is not fundamental and time is an induced property of slow moving systems.

 

 

Ref: On Geometrogenesis (the only thing I would disagree with this blog, is the contention ''Geometrogenesis'' was first used in Markoupoulou's work. This is not true, it was introduced by Wheeler) http://guidetoreality.blogspot.co.uk/2007/04/geometrogenesis.html

Edited November 12, 2012 by Aethelwulf

[Aethelwulf]

Why can't we determine an energy for the universe when it came into existence?

 

 

 

Well, I will explain this from a geometrical then a quantum stance. Heisenberg uncertainty is a form of the geometric Cauchy Schwarz inequality law and this might be a clue to how to treat the universe when it first came into existence. The reason why, is because the geometric Cauchy Schwarz inequality states that you can deal with triangles in spacetime on the fundamental level, and their sides reveal an uncertainty relationship - [math]a[/math] must be less than or equal to [math]b+c[/math], [math]b[/math] less than or equal to [math]a+c[/math], and [math]c[/math] must be less than or equal to [math]a+b[/math]. Attempting to violate this leads to uncertainties in your measurements, just like what you would obtain from Heisenberg's Uncertainty Principle in it's full form.

 

Therefore, if you wind the clock back to the first instant, we don't have any lengths in our triangle, owing to a very large uncertainty in our energy. When the observable universe is squeezed into a small area, the space it occupies becomes extremely uncertain. Say we could try and measure this between the scales of [math]10^{−37}[/math] and [math]10^{−10}[/math] of a second, this is very small time-wise, but because of the relationship [math]\Delta E \Delta t[/math] we run into problems inherent with the uncertainty principle. The problem is, is that it is far too small to actually measure a well-defined energy!7

 

So not only can we argue, the universe cannot have a well-defined energy because no one has observed it... but we can also argue the Uncertainty Principle wages problems with well-defined quantities of energy as well.

[Aethelwulf]

The fact energy was undefined, could give possible pathways to explain why the universe is now accelerating in it's expansion. To explain this better, the universe could only expand steadily if there was an equal of distribution of energy at any ''time'' given. However, there seems to be an exponential increase of energy and we have evidence suggesting that energy is not conserved on the cosmological scale. If energy is not conserved, we should start thinking in these quantum terms where energy is highly undefined.

[Aethelwulf]

How is time a property of slow moving systems?

 

 

It has to do with relativity again. Bradyons, or Tardyons, are slow matter moving systems capable of measuring relativistic clock dynamics. We are bradyons. We are not relativistic, those particles which are relativistic have a lorentz transformation which permits time to not pass at all. For photons, they do not even possess a frame of reference, which is an interesting anomaly we have came to realize, because things travelling at the speed of light, don't even experience a single chronon pass.

[Aethelwulf]

A chronon by the way, is a measurement of time. A short one, but not the shortest theoretically-known.

[Aethelwulf]

Here is a paper explaining how spacetime is emergent... in my words, but equal, is induced time, space is emergent to matter, time is induced by systems moving in the low energy phase but equally able to comprehend it.

 

 

http://arxiv.org/pdf/1206.0085.pdf

[Aethelwulf]

The Daily Galaxy has reported another breakthrough, that spacetime has no time dimension, http://www.dailygalaxy.com/my_weblog/2011/04/spacetime-has-no-time-dimension-new-theory-claims-that-time-is-not-the-4th-dimension.html

 

 

The next move from this, is the realization that space does not exist fundamentally. In Barbour's work, he explained how there is no background space, only relative objects make up the background of space.

[Aethelwulf]

This paper, finally reduces to my own speculations, posted on the gravity institute of investigations, FQXI

 

http://fqxi.org/data/essay-contest-files/Fink_Speculation_Regarding.pdf?phpMyAdmin=0c371ccdae9b5ff3071bae814fb4f9e9

[Aethelwulf]

Time is completely separate to space. Even Dirac himself noted that if he had a choice, he would say that time is not fundamental from the Minkowski set-up of space and time being united. If time is not fundamental, we must begin to question space as well, especially in the primal existence of this universe as we have been notoriously been led to believe.

[Aethelwulf]

Here is a nice professional paper on the possibility that spacetime is emergent

 

http://arxiv.org/abs/0711.4416