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## Trading in the Real Time for Real Space

Note for Reader: Real Space is Imaginary Time while Imaginary Space is Real Time.

Trading in the Real Time for Real Space

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:

"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 , 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 and the time , 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 -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 . 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.

[b]Conclusions[b]

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/reali...ler_dewitt.asp

[2] http://arxiv.org/PS_cache/gr-qc/pdf/9812/9812027v1.pdf

''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

*Note
I no longer believe the Wick Rotation will act as a solution. Instead, it involves two reference frames of time which will somehow, I suspect be integrated into the derivations which lead to the Wheeler de Witt description, leaving an imaginary time description.
Last edited by Green Destiny; 2010-Nov-08 at 12:18 AM.

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Mmmm... sorry. I will fix the latex, must be the wrong code. brb

$1$

$1$

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Oh this place doesn't use latex then?

4. Nope, no latex here but it isn't hard to use another latex from some other site here

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Originally Posted by Green Destiny

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 . 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:
Q1: Really? And how does the above change of coordinate retain the metric invariance?

Q2: Can you still prove the Lorentz covariance in after the change of coordinates?Something simple, like the Maxwell wave equation in vacuum? Please, give it a try.

6. Here is a link to a handy LaTex to .gif converter: http://www.codecogs.com/latex/eqneditor.php

7. Ugh... tried to post the link instead of the .gif. This is what the output from the Latex to gif looks like.

Sorry for the double post.

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Originally Posted by Green Destiny
Note for Reader: Real Space is Imaginary Time while Imaginary Space is Real Time.

Trading in the Real Time for Real Space

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:

"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 , so an action can be given as:

where

The corresponding momenta can be given when knowing that our new langrangian depends on two coordinates and the time , and with Eulers theorem, one finds that the Canonical Hamiltonian vanishes:

where the momentum is:

and

The quantum cosmological approach will have an equation describing this as:

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 -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

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 . 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.

[b]Conclusions[b]

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/reali...ler_dewitt.asp

[2] http://arxiv.org/PS_cache/gr-qc/pdf/9812/9812027v1.pdf

''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

*Note
I no longer believe the Wick Rotation will act as a solution. Instead, it involves two reference frames of time which will somehow, I suspect be integrated into the derivations which lead to the Wheeler de Witt description, leaving an imaginary time description.

look , both real time and real space are the consequence of , not before , energy and /or matter

real space is a consequece of energy/matter

real time only comes into the equation when energy/matter interact with each other

understand my point ? people ?

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what I'm getting at , is that , there is a difference between space and time

space is space because of energy/matter

time is not real , it is a consequence of, ... energy/matter=space or space=energy/matter

if time is a real physical entity , then let time and time alone prove that it has the physical substance to influence any world

10. Originally Posted by Green Destiny
Oh this place doesn't use latex then?

11. Originally Posted by north
what I'm getting at , is that , there is a difference between space and time

space is space because of energy/matter

time is not real , it is a consequence of, ... energy/matter=space or space=energy/matter

if time is a real physical entity , then let time and time alone prove that it has the physical substance to influence any world

No, even this explanation does not make any sense.

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Originally Posted by macaw
Q1: Really? And how does the above change of coordinate retain the metric invariance?

Q2: Can you still prove the Lorentz covariance in after the change of coordinates?Something simple, like the Maxwell wave equation in vacuum? Please, give it a try.
No, because tranformations as such fail. Indeed, this is a classical feature of GR since diffeomorphism invariance leads to non-existent moving frames. To remove this problem, I have adopted a new course of action. I'm afraid moving frames of references in terms of Lorentz Covariance works neither way, but that was not an issue of the OP, so I see no reason further discussing it.

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Originally Posted by Green Destiny
No, because tranformations as such fail.
Q3: So, you admit that your transformations do not preserve metric invariance?

Indeed, this is a classical feature of GR since diffeomorphism invariance leads to non-existent moving frames.
Q4: Can you prove that, please. With math, not with buzzwords.

To remove this problem, I have adopted a new course of action.

I'm afraid moving frames of references in terms of Lorentz Covariance works neither way,
Q6: Really? How come it works in all tests of relativity?

but that was not an issue of the OP, so I see no reason further discussing it.
Well, I see a very good reason, so please stop dodging and start answering the questions.

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First of all I am not dodging anything. You asked about metric invariance - I told you straight out this is not as issue any more than finding out we have a static time frame on a cosmological outlook from the WDW equation. If you don't like that answer, tough. My model was not there to preserve that quality, nor is it there to address that issue. Moving frames of references fail in WDW equation so such frames related to metric invariance pretty much is a seperate issue, which my OP does not challenge or address, you are causing an unecessery topic of discussion.

Q4: Can you prove that, please. With math, not with buzzwords.

I have in the OP, if you had not noticed. The classical Langrangian which leads to a constraint on the Hamiltonian which yields . Because general relativity is diffeomor-
phism invariant and because time evolution is nothing but a special type of diffeomorphism the
Hamiltonian of the classical theory is nothing but a constraint. How did you not notice this? Or are you just trying to stab questions where not necessery like I've seen so far?

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Also, You are asking how it works in tests of relativity. You are forgetting the Hamiltonian constraint and the Timelessness and ceasing of reference frames are a paradox of relativity and are usually simply ignored or never tackled. If you asked someone to theoretically prove timelessness and ceasing of reference frames, then of course we will resort to a model we understand and can readily use, hence why such models prevail. Your questions are filled with misunderstanding.

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Sorry, you wanted the formalism as well, where I say I have taken a new course of action?

[LaTeX ERROR: Convert failed]

where here is our matter field in minisuperspace. The equation can be solved by a seperation of variables to give two coupled equations:

The definition of time now can be given as one of two solutions - to describe our time, we have also two choices, we can define our time as either the scale factor or as the matter field which I have represented with a different description to identify the two - these two different descriptions could be given a unique transformation. We can view one trivially as an imaginary time dimension, by a wick rotation, and one as a real time, given by our Hamiltonian.

But the real question is which time reference do we make real and which one imaginary? Interestingly, we would run into all sorts of problems if we performed the wick rotation on the matter field, namely, imaginary mass descriptions. Performing the wick rotation on the scale factor will rid us of our matter field because real time calculations would have vanish due to the WDW equation. Though, we can simply say ''well, no use with that any more'' and resort to the final solution of an imaginary time reference on the theory.

http://arxiv.org/PS_cache/hep-th/pdf/9503/9503073v2.pdf

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Originally Posted by Green Destiny
First of all I am not dodging anything. You asked about metric invariance - I told you straight out this is not as issue any more than finding out we have a static time frame
Q7: What is a "static time frame"? Define it , please?

on a cosmological outlook from the WDW equation.
Q8: What is "cosmological outlook"?

Q9: What is the "WDW" equation?

Q10: Can you write down the above in a mathematical formalism, please?

Q11 Can you prove your above assertion? Math please, no string of buzzwords.

If you don't like that answer, tough. My model was not there to preserve that quality,
Q12 So, your theory does not preserve metric invariance? This is the third time I am asking you this direct question. Yes or no?

nor is it there to address that issue.
Q13: Why not? All mainstream theories do.

Moving frames of references fail in WDW equation so such frames related to metric invariance pretty much is a seperate issue, which my OP does not challenge or address, you are causing an unecessery topic of discussion.
No, I am asking you direct questions that you keep trying to dodge. So, once again:

Q14: Can you demonstrate the invariance of the Maxwell equations in your formalism? Yes or no?

Q4: Can you prove that, please. With math, not with buzzwords.

I have in the OP, if you had not noticed.
No, you haven't. Please stop dodging and do it.

The classical Langrangian which leads to a constraint on the Hamiltonian which yields . Because general relativity is diffeomor-
phism invariant and because time evolution is nothing but a special type of diffeomorphism the
Hamiltonian of the classical theory is nothing but a constraint.
Please stop stringing meaningless buzzwords and answer the exact questions I've asked you. The above has nothing to do with the question you've asked. Do you know how to answer the question? Yes or no?

How did you not notice this? Or are you just trying to stab questions where not necessery like I've seen so far?
Last edited by macaw; 2010-Nov-19 at 03:09 AM.

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If you truely think my statement on diffeomorphism invariance and classical hamiltonian contraints was rubbish, or useless, then you are wasting my time. Especially when you are asking me to write the WDW equation, and its written in the OP.

My session with you is being terminated.

19. Let's tone down the animus, please.

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Originally Posted by Green Destiny
If you truely think my statement on diffeomorphism invariance and classical hamiltonian contraints was rubbish, or useless,

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Originally Posted by Green Destiny

$I= \int L(q, \dot{q})= \int d\tau L_{\tau}$

where $L_{\tau}= q_{0}_{\tau} L(q,q_{\tau}/q_{0}_{\tau})$
Let's stop right here, the above formula is pure nonsense. This shows that , though you copied it from a book on the internet, you do not understand what you are copying and pasting. Not only that the formula is grossly incorrect but you clearly do not understand what it means.

where the momentum is:

$p_0= \frac{\partial L}{\partial q_{0}_{\tau}}$

and

$p= \frac{\partial L}{\partial q_{\tau}}$
This is in the "not even wrong" category. Both expressions are incorrect. This is basic variational calculus, the above are not the correct definitions of the variables for calculating the Lagrangian. This is obvious to anyone that has basic knowledge of variational mechanics.

The quantum cosmological approach will have an equation describing this as:

$\hat{H}\Psi= (\frac{2\pi G \hbar^2}{3} \frac{\partial^2}{\partial \alpha^2})+\sum_i[-\frac{\hbar^2}{2} \frac{\partial^2}{\partial x_{i}^{2}}+V_i(\alpha,x_i}]) = 0$
The above is not an equation because it is missing a very important piece. Out of curiosity:

Q15: Do you know what is missing?
Q16: Can you repair the above such that it has meaning?
Q17: What type of equation would you get if you managed to write it correctly?
Last edited by macaw; 2010-Nov-19 at 03:57 AM.

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I do not believe that first equation has came out correctly due to latexing here, but either way, the Langrangian is perfectly true. There are similar derivations, and what you see above is one of the standard derivisions. However, the last equation is missing a psi, just a typo made a while back and I never got it fixed. I apologize for that.

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brb

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Yes.. I doubt the first equations are wrong, they were information extracted from this source, which was itself taken from a physics textbook

http://www.ipod.org.uk/reality/reali...ler_dewitt.asp

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And fancy that, I look for more links which could back that claim up above, and behold http://galileo.phys.virginia.edu/cla...neticField.htm ~ yes, there truely is nothing wrong with those equations. Would you like to divulge what you thought was wrong with the Canonical Momentum?

(ps) I cannot add the psi yet. I've not learned the latexing system here, or how it is done.

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So now let us see if we can find that action somewhere, and put that to rest. It be of no surprise I did not have to look very far http://en.wikipedia.org/wiki/Action_(physics) The ''I'' in my equation is also a symbol used for action, as it is also denoted with the letter ''S''. In the wiki article, action is given as S= \int L(x, dot{x}, t)dt - The action I gave is L(x dot{x}) - anyone with any mathematical training will know instinctively there is nothing wrong with my expression. It is the Langrangian, which is why.

Like to tell me what was wrong with it? This is a recurring theme with you now. I don't even think you know what you are objecting to.

27. No you forgot a Ψ in the rhs of the equation HΨ = .....

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I know. It was pointed out to me above. I am not sure how to correct the latexing here, which is why I haven't fixed it.

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But the other equations are fine. the Canonical momentum is correct and so is the Langrangian.

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Originally Posted by Green Destiny
I do not believe that first equation has came out correctly due to latexing here, but either way, the Langrangian is perfectly true. There are similar derivations, and what you see above is one of the standard derivisions. However, the last equation is missing a psi, just a typo made a while back and I never got it fixed. I apologize for that.
ALL your equations have come out incorrect, proving that you do not understand ANY of them. This is not a latex problem, this is clearly your lack of understanding of what they really mean.To further prove that this is the case, here are some additional questions:

Take the Minkowski metric: ds^2=(cdt)^2-dx^2
Q16. Find the Lagrangian associated with this metric
Q17. Find the momenta from the Lagrangian
Q18. Write down the equations of motion
Q19. Solve the equations of motion