Note for Reader: Real Space is Imaginary Time while Imaginary Space is Real Time.
Trading in the Real Space for Real Time
In this essay, I will attempt to put forward an idea which will describe the universe from two different perspectives involving time, and how both perspectives can work for each other in eliminating the no-time problem of Quantum Mechanics.
There is no Arrow of Time
I have independantly came to the conclusion in the past to myself, that an arrow of time cannot exist. This depends on two things:
A) To define some definate arrow from the past into the future, there needs to be a point in where everything came from to define some direction in space. Direction does not exist in space. Equally there is no center to the universe according to current understanding, every point on the spacetime map would be the center to the universe.
B) Time does not have a flow.
The last concept I found out was used in a strong arguement against the arrow of time:
http://www.motionmountain.net/download.html
"Time is a concept introduced specially to describe the flow of events around us; it does not itself flow, it describes flow. Time does not advance. Time is neither linear nor cyclic. The idea that time flows is as hindering to understanding nature as is the idea that mirrors Page 71 exchange right and left. The misleading use of the expression ‘flow of time’, propagated first by some flawed Ref. 36 Greek thinkers and then again by Newton, continues. Aristotle (384/3–322 bce), careful to think logically, pointed out its misconception, and many did so after him. Nevertheless, expressions such as ‘time reversal’, the ‘irreversibility of time’, and the much-abused ‘time’s arrow’ are still common. Just read a popular science magazine chosen at random.
Without an arrow of time, the view where time rushed unequivocably from past to the future, while we are inexorably stuck in the present. But current view attends the idea that the past doesn't even exist, but only as a record of the present, a view held strongly by Dr Wheeler. In a sense, it exists itself an arguement based on a deep technicality of ''when things exist''. Things, that is, events themselves only ever happen inside the present sphere. The future cones and past cones exist with their own present frames, and notions of an absolute past and an absolute future cease to exist.
The Wheeler de Witt Equation
The problem with covariance in a quantum cosmological look is that gravity is described by diffeomorphism invariance contraint on the equations describing the evolution of the universe as a whole. This quantum cosmological approach has been widely accepted as one of main problems dealing with an approach towards unified theory on gravity.
A way to derive the equation other than quantizing Einsteins equation comes in a derivation. We must first consider a langrangian $$L(q, \dot{q})$$ where it is said the time parameter defines a specific frame of reference and one can define [1] a new time as a monotonic function of $$t$$, so an action can be given as:
$$I= \int L(q, \dot{q})= \int d\tau L_{\tau}$$
where $$L_{\tau}= q_{0}_{\tau} L(q,q_{\tau}/q_{0}_{\tau})$$
The corresponding momenta can be given when knowing that our new langrangian depends on two coordinates $$(q,q_0)$$ and the time $$\tau$$, and with Eulers theorem, one finds that the Canonical Hamiltonian vanishes:
$$\mathbb{H}= \psi(p_0 q_{0}_{\tau}+p q_{\tau}-L)= 0$$
where the momentum is:
$$p_0= \frac{\partial L}{\partial q_{0}_{\tau}}$$
and
$$p= \frac{\partial L}{\partial q_{\tau}}$$
The quantum cosmological approach will have an equation describing this as:
$$\hat{H}\Psi= (\frac{2\pi G \hbar^2}{3} \frac{\partial^2}{\partial \alpha^2})+\sum_i[-\frac{\hbar^2}{2} \frac{\partial^2}{\partial x_{i}^{2}}+V_i(\alpha,x_i}]) = 0$$
And basically, the zero quantity refers to time, and mathematically it has plagued the minds of physicists, asking exactly how should we deal with this bizarre nature?
From now on, my approach will be to treat the time evolution in the Hamiltonian as a real time coordinate. The importance in desctinguishing this is to assertain two descriptions of time, and in doing so, will allow me to hypothesize an understanding on the real time measurements in the Hamiltonian of the Wheeler de Witt equation and one in understanding a comsological view of the universe concerning the beginning of time using imaginary time coordinates.
All in all, as Einstein similarily said, for those of us who believe in quantum theory, we must believe that the destinction of past and future are only stubborn illusions; either this, or our understanding of the $$\psi$$-function is incomplete, just as all the information in the Hamiltonian is encoded in the wave function of the universe.
There was no Beginning
According to Stephen Hawking the universe can be viewed to not have a beginning [2], if one looks at the universe in terms of purely imaginary time. Real time events would be when the universe could be said to have just began, but in imaginary reality of time, the big bang vanishes to exist. In his idea, real time is only a tool used by us to describe events in the universe. Similar to that idea, the real time evolution in the Hamiltonian of the Wheeler de Witt equation no longer has a measurable quantity because of the problem of diffeormorphism Invariances on the theory, so from this angle, we may actually forget about real time measurements, and trade it for all it's worth.
By keeping the real time description provided by the Wheeler de Witt equation can we preserve Hawking's vision of a universe in imaginary time. Before this realization, Hawking would have had to admit that the big bang was not exactly made to ''disappear'' - he stresses the universe still has a beginning, only that this view becomes distorted under the correct light. Now we can maintain that view, by adopting that there are no real time measurements at all. We are basically trading in the paradox of time, for a new one, one which has serious implications on the bizarre nature of time and of course, the continuing question of whether a big bang truely existed. By allowing imaginary time into the equations, one can use the Wheeler de Witt equation therefore to justify why we should not take real time seriously.
If we remove the singularity and the beginning of time with imaginary time (real space) we can throw out the paradox of real time remaining by introducing the firm belief in the Wheeler de Witt equation - this requires two different kind of times.
Imaginary Time Evolution
Imaginary time evolution is given as $$-\psi_{\alpha}= (\omega + E - H)^2 \psi$$
Real time extends to our (past) and our (future) - imaginary time is vertical or at 90 degrees off the real time. A Wick Rotation allows us to transform real time into imaginary time, so it may be interpreted that we can easily change our coordinate system in terms of $$t \rightarrow i\tau$$. The proceedure actually changes the Lorentzian metric $$(-+++)$$ to an euclidean metric of signature $$(++++)$$ and is said to be a good way to go down because:
(1) it makes the path integral solvable
(2) the procedure can be used to explain the thermodynamic properties of black holes
(3) it makes it possible to measure gravitational instantons
So even if changing real time into imaginary time is just a mathematical trick, it has logical applications. In order however to take the stance we have in this essay, we must view the Hamiltonian of the Wheeler de Witt equation as a non-physical application on the theory, and begin to view imaginary time as the only true definition of time in our universe. Essentially, real time vanishes, while imaginary prevails.
Doing this solves:
(1) The problem of the beginning of time including the singularity
(2) The arrow of time, because real time is only an illusion
(3) successfully allows us to remove real time for imaginary time (Solving the no-time problem in GR)
The only way to make the idea successful is to find a way to allow the Wheeler de Witt equation to mathematically co-exist with imaginary time, or the no-boundary proposal. So far, the only clear way to do this, is to find a way to describe the universes wave function in terms of two time perspectives, which are fundamentally independant of each other, those being dressed in real time, and those dressed in imaginary time.
reflection
I forwarded the idea to a PhD because I felt it was a promising approach. The doctor replied saying ''it's a very good idea, but I doubt the Wick Rotation will act as a solution for the theory.'' So reflecting on this, I no longer believe the Wick Rotation will act like a solution to Real Time vs Real Space concept.
Conclusions
There is no explicit time description of the Hamiltonian in the Canonical Wheeler de Witt equation. The time evolution being described may very well describe real time events. By treating it as such, it seems other points of view on time can exist, views which themselves seem to solve the problem of whether a beginning of time should be viewed at all.
[1] http://www.ipod.org.uk/reality/reality_wheeler_dewitt.asp
[2] http://arxiv.org/PS_cache/gr-qc/pdf/9812/9812027v1.pdf
Further reading
http://books.google.co.uk/books?id=...esnum=8&ved=0CEoQ6AEwBzgK#v=onepage&q&f=false
''an incomplete wave function could be possible describing the universe''
~ on the wave function http://arxiv.org/PS_cache/gr-qc/pdf/0308/0308029v1.pdf
http://arxiv.org/PS_cache/gr-qc/pdf/0308/0308029v1.pdf
http://en.wikipedia.org/wiki/Imaginary_time
Timelessness in GR - http://www.fqxi.org/community/forum/topic/376
The Flow of Time*
By George F. R. Ellis - http://www.fqxi.org/community/essay/winners/2008.1
Trading in the Real Space for Real Time
In this essay, I will attempt to put forward an idea which will describe the universe from two different perspectives involving time, and how both perspectives can work for each other in eliminating the no-time problem of Quantum Mechanics.
There is no Arrow of Time
I have independantly came to the conclusion in the past to myself, that an arrow of time cannot exist. This depends on two things:
A) To define some definate arrow from the past into the future, there needs to be a point in where everything came from to define some direction in space. Direction does not exist in space. Equally there is no center to the universe according to current understanding, every point on the spacetime map would be the center to the universe.
B) Time does not have a flow.
The last concept I found out was used in a strong arguement against the arrow of time:
http://www.motionmountain.net/download.html
"Time is a concept introduced specially to describe the flow of events around us; it does not itself flow, it describes flow. Time does not advance. Time is neither linear nor cyclic. The idea that time flows is as hindering to understanding nature as is the idea that mirrors Page 71 exchange right and left. The misleading use of the expression ‘flow of time’, propagated first by some flawed Ref. 36 Greek thinkers and then again by Newton, continues. Aristotle (384/3–322 bce), careful to think logically, pointed out its misconception, and many did so after him. Nevertheless, expressions such as ‘time reversal’, the ‘irreversibility of time’, and the much-abused ‘time’s arrow’ are still common. Just read a popular science magazine chosen at random.
Without an arrow of time, the view where time rushed unequivocably from past to the future, while we are inexorably stuck in the present. But current view attends the idea that the past doesn't even exist, but only as a record of the present, a view held strongly by Dr Wheeler. In a sense, it exists itself an arguement based on a deep technicality of ''when things exist''. Things, that is, events themselves only ever happen inside the present sphere. The future cones and past cones exist with their own present frames, and notions of an absolute past and an absolute future cease to exist.
The Wheeler de Witt Equation
The problem with covariance in a quantum cosmological look is that gravity is described by diffeomorphism invariance contraint on the equations describing the evolution of the universe as a whole. This quantum cosmological approach has been widely accepted as one of main problems dealing with an approach towards unified theory on gravity.
A way to derive the equation other than quantizing Einsteins equation comes in a derivation. We must first consider a langrangian $$L(q, \dot{q})$$ where it is said the time parameter defines a specific frame of reference and one can define [1] a new time as a monotonic function of $$t$$, so an action can be given as:
$$I= \int L(q, \dot{q})= \int d\tau L_{\tau}$$
where $$L_{\tau}= q_{0}_{\tau} L(q,q_{\tau}/q_{0}_{\tau})$$
The corresponding momenta can be given when knowing that our new langrangian depends on two coordinates $$(q,q_0)$$ and the time $$\tau$$, and with Eulers theorem, one finds that the Canonical Hamiltonian vanishes:
$$\mathbb{H}= \psi(p_0 q_{0}_{\tau}+p q_{\tau}-L)= 0$$
where the momentum is:
$$p_0= \frac{\partial L}{\partial q_{0}_{\tau}}$$
and
$$p= \frac{\partial L}{\partial q_{\tau}}$$
The quantum cosmological approach will have an equation describing this as:
$$\hat{H}\Psi= (\frac{2\pi G \hbar^2}{3} \frac{\partial^2}{\partial \alpha^2})+\sum_i[-\frac{\hbar^2}{2} \frac{\partial^2}{\partial x_{i}^{2}}+V_i(\alpha,x_i}]) = 0$$
And basically, the zero quantity refers to time, and mathematically it has plagued the minds of physicists, asking exactly how should we deal with this bizarre nature?
From now on, my approach will be to treat the time evolution in the Hamiltonian as a real time coordinate. The importance in desctinguishing this is to assertain two descriptions of time, and in doing so, will allow me to hypothesize an understanding on the real time measurements in the Hamiltonian of the Wheeler de Witt equation and one in understanding a comsological view of the universe concerning the beginning of time using imaginary time coordinates.
All in all, as Einstein similarily said, for those of us who believe in quantum theory, we must believe that the destinction of past and future are only stubborn illusions; either this, or our understanding of the $$\psi$$-function is incomplete, just as all the information in the Hamiltonian is encoded in the wave function of the universe.
There was no Beginning
According to Stephen Hawking the universe can be viewed to not have a beginning [2], if one looks at the universe in terms of purely imaginary time. Real time events would be when the universe could be said to have just began, but in imaginary reality of time, the big bang vanishes to exist. In his idea, real time is only a tool used by us to describe events in the universe. Similar to that idea, the real time evolution in the Hamiltonian of the Wheeler de Witt equation no longer has a measurable quantity because of the problem of diffeormorphism Invariances on the theory, so from this angle, we may actually forget about real time measurements, and trade it for all it's worth.
By keeping the real time description provided by the Wheeler de Witt equation can we preserve Hawking's vision of a universe in imaginary time. Before this realization, Hawking would have had to admit that the big bang was not exactly made to ''disappear'' - he stresses the universe still has a beginning, only that this view becomes distorted under the correct light. Now we can maintain that view, by adopting that there are no real time measurements at all. We are basically trading in the paradox of time, for a new one, one which has serious implications on the bizarre nature of time and of course, the continuing question of whether a big bang truely existed. By allowing imaginary time into the equations, one can use the Wheeler de Witt equation therefore to justify why we should not take real time seriously.
If we remove the singularity and the beginning of time with imaginary time (real space) we can throw out the paradox of real time remaining by introducing the firm belief in the Wheeler de Witt equation - this requires two different kind of times.
Imaginary Time Evolution
Imaginary time evolution is given as $$-\psi_{\alpha}= (\omega + E - H)^2 \psi$$
Real time extends to our (past) and our (future) - imaginary time is vertical or at 90 degrees off the real time. A Wick Rotation allows us to transform real time into imaginary time, so it may be interpreted that we can easily change our coordinate system in terms of $$t \rightarrow i\tau$$. The proceedure actually changes the Lorentzian metric $$(-+++)$$ to an euclidean metric of signature $$(++++)$$ and is said to be a good way to go down because:
(1) it makes the path integral solvable
(2) the procedure can be used to explain the thermodynamic properties of black holes
(3) it makes it possible to measure gravitational instantons
So even if changing real time into imaginary time is just a mathematical trick, it has logical applications. In order however to take the stance we have in this essay, we must view the Hamiltonian of the Wheeler de Witt equation as a non-physical application on the theory, and begin to view imaginary time as the only true definition of time in our universe. Essentially, real time vanishes, while imaginary prevails.
Doing this solves:
(1) The problem of the beginning of time including the singularity
(2) The arrow of time, because real time is only an illusion
(3) successfully allows us to remove real time for imaginary time (Solving the no-time problem in GR)
The only way to make the idea successful is to find a way to allow the Wheeler de Witt equation to mathematically co-exist with imaginary time, or the no-boundary proposal. So far, the only clear way to do this, is to find a way to describe the universes wave function in terms of two time perspectives, which are fundamentally independant of each other, those being dressed in real time, and those dressed in imaginary time.
reflection
I forwarded the idea to a PhD because I felt it was a promising approach. The doctor replied saying ''it's a very good idea, but I doubt the Wick Rotation will act as a solution for the theory.'' So reflecting on this, I no longer believe the Wick Rotation will act like a solution to Real Time vs Real Space concept.
Conclusions
There is no explicit time description of the Hamiltonian in the Canonical Wheeler de Witt equation. The time evolution being described may very well describe real time events. By treating it as such, it seems other points of view on time can exist, views which themselves seem to solve the problem of whether a beginning of time should be viewed at all.
[1] http://www.ipod.org.uk/reality/reality_wheeler_dewitt.asp
[2] http://arxiv.org/PS_cache/gr-qc/pdf/9812/9812027v1.pdf
Further reading
http://books.google.co.uk/books?id=...esnum=8&ved=0CEoQ6AEwBzgK#v=onepage&q&f=false
''an incomplete wave function could be possible describing the universe''
~ on the wave function http://arxiv.org/PS_cache/gr-qc/pdf/0308/0308029v1.pdf
http://arxiv.org/PS_cache/gr-qc/pdf/0308/0308029v1.pdf
http://en.wikipedia.org/wiki/Imaginary_time
Timelessness in GR - http://www.fqxi.org/community/forum/topic/376
The Flow of Time*
By George F. R. Ellis - http://www.fqxi.org/community/essay/winners/2008.1