Is There Physical Significance to Imaginary Dimensions in Space-Time?

[Anonym]

However, it is less known that quantum mechanics can be formulated in a completely different, but equivalent language. It is called "quantum logic", or theory of "orthomodular lattices", which is also related to "projective geometry". In this approach one deals with probabilities rather than with probability amplitudes. So, complex numbers never appear.

The language of orthomodular lattices is exactly isomorphic to the language of subspaces in the Hilbert space (Piron's theorem), and, in principle, it allows to perform all quantum mechanical calculations. However, lattice theory is an obscure (from the point of view of physicists) corner of mathematics. So, physicists haven't developed intuition to deal with lattices, even remotely comparable with intuition about vector spaces.

You describe the same “obscure (from the point of view of physicists) corner of mathematics” and mention the results obtained by the same “company” of physicists that we discuss with Mentz114, namely, E.P. Wigner, J. von Neumann, P. Jordan, G. Birkhoff, E.C.G. Stueckelberg, M. Guenin, H.Ruegg, C. Piron, G.W. Mackey, A.M. Gleason, J.M. Jauch, G.G. Emch, D. Finkelstein, D. Speiser and more recently L.P. Horwitz (my spiritual “father” during Ph.D. studies), S.L. Adler, A. Zeilinger.

You are talking about the “R-process” and not “U-process” in R. Penrose terminology. Your post is connected to the Theory of Measurements which indeed very interesting and important. However, it was demonstrated by A. Einstein (5th Solvay) that the “R-process” is instant, delta(t)=0. Therefore, it is apparently not connected with the nature of time.

Regards, Dany.

P.S. By the way, all that is “Vanilla quantum mechanics”. That “mysterious” R-process of duration zero provides also the available time interval for realization of M.Born statistical interpretation of Quantum Physics.

 

[Anonym]

My intuition says the opposite - but I'm not sure either. I thought of the Pauli and Dirac algebras, but now I'm not sure if they can be represented by matrices over real numbers only. Isn't the chiral representation of the Dirac algebra real ?

I don't think that i is essential to the Schrodinger equation which can be decomposed into two real equations. But you'll no doubt disagree with that.

Indeed I disagree with that (but I am not sure either). You point out the connection between the two-component wave packet and two-level physical system (see E.C.G. Stueckelberg et al, A. Zeilinger et al). However if you write 2x2 matrix with real matrix elements instead of i you do nothing, it is matter of notation only. For me personally the 2x2 form provides more clear physical content.

I consider the original Schrödinger equation as the QM analog of the classical static. Time there is not a time. The equation has only stationary solutions. However, if you write the same Schrödinger equation for two-level system (2x2) with the complex matrix elements and will maintain i as in the original Schrödinger paper, the equation turns out now to be dynamical and the same t will be now time. Indeed that trick breaks completely the mathematical structure of Hilbert space based solely on the 2-dim quadratic normal division algebra of complex numbers.

Perhaps, the original Schrödinger equation is the static limit of that “Schrödinger equation” (see M.P. Silverman, “More Than One Mystery”, Springer-Verlag,NY, 1995 (p.146, eq. 23(a) or something close to it). Notice that 2x2 matrices with complex matrix elements may be written in the basis of 4-dim quadratic normal division algebra of Hamilton quaternions (with the signature {+,-,-,-}). I should find it connection with the equations of motion for the fundamental fermions.

Regards, Dany.

P.S. The Pauli and Dirac matrices (C2 and C4 respectively) always may be written using real matrix elements since the Clifford algebras are associative.

 

[meopemuk]

You describe the same “obscure (from the point of view of physicists) corner of mathematics” and mention the results obtained by the same “company” of physicists that we discuss with Mentz114, namely, E.P. Wigner, J. von Neumann, P. Jordan, G. Birkhoff, E.C.G. Stueckelberg, M. Guenin, H.Ruegg, C. Piron, G.W. Mackey, A.M. Gleason, J.M. Jauch, G.G. Emch, D. Finkelstein, D. Speiser and more recently L.P. Horwitz (my spiritual “father” during Ph.D. studies), S.L. Adler, A. Zeilinger.

That's an impressive list of names. My own first attempts to go beyond college physics were associated with reading a series of papers by Horwitz and Biedenharn on "octonionic quantum mechanics". Then I found (I think it was a footnote) that this approach violates Mackey axioms. That's how I learned about works of Birkhoff, von Neumann, Piron, and the rest of the crowd.

You are talking about the “R-process” and not “U-process” in R. Penrose terminology. Your post is connected to the Theory of Measurements which indeed very interesting and important. However, it was demonstrated by A. Einstein (5th Solvay) that the “R-process” is instant, delta(t)=0. Therefore, it is apparently not connected with the nature of time.

I am not familiar with "U" and "R" terminology. If I guessed, I would say that U is unitary time evolution, and R is wavepacket "reduction". Am I right?

 

[Mentz114]

Hi Dany:

You point out the connection between the two-component wave packet and two-level physical system (see E.C.G. Stueckelberg et al, A. Zeilinger et al)

I meant that the Schrodinger equation can be written as a real part, and an imaginary part - hence two equations. Not quite as sublime as your interpretation of my remark.
Thanks for pointing out that C2 and C4 are the Pauli and Dirac algebras. I once knew that !

If you're interested in quaternions in physics, have a look at Doug Sweetser's site
www.quaternions.com

 

[Anonym]

I am not familiar with "U" and "R" terminology. If I guessed, I would say that U is unitary time evolution, and R is wavepacket "reduction". Am I right?

Yes.

That's how I learned about works of Birkhoff, von Neumann, Piron, and the rest of the crowd.

I like your metaphors - “Vanilla” (hadrons, white holes, etc) and “obscure corner” (Castalia around Lake Geneva). And it was Larry’s dream Saas-Fee on top.

Perhaps there you may grasp nature of time.

Regards, Dany.

 

[SpitfireAce]

Who taught you that? If you will read A. Einstein original paper Ann. Phys., 17, 891, 1905 (Ch.1, Kinematical Part, par. 1-5) you will discover that A. Einstein was not less smart than you and did exactly what you consider the natural physical approach.

Einstein titled that paper "On the Electrodynamics of Moving Bodies"

In part one he defines simultaneity for stationary bodies... then in the very beginning of part 2 ("On the Relativity of Lengths and Times") before he even starts speaking about moving bodies he writes:

"The following reflexions are based on the principle of relativity and on the
principle of the constancy of the velocity of light."

it is more than obvious that he derived SR from the law of propagation, it is the whole foundation for Lorentz transformation... that a stationary body and a moving body should measure the same speed for light

I think you misunderstood me

 

[Anonym]

Einstein titled that paper "On the Electrodynamics of Moving Bodies"

In part one he defines simultaneity for stationary bodies... then in the very beginning of part 2 ("On the Relativity of Lengths and Times") before he even starts speaking about moving bodies he writes:

"The following reflexions are based on the principle of relativity and on the
principle of the constancy of the velocity of light."

it is more than obvious that he derived SR from the law of propagation. I think you misunderstood me

No. You misunderstood him. The law of propagation is Maxwell equations. It is more than obvious that you confuse kinematics and dynamics. It is the standard “vanilla” way to formulate the theory. P.A.M. Dirac (for example) in “Principles of QM” did exactly the same (Ch.1-4 and the rest).

A.Einstein used word “light” instead of “signal” since it was the problem in front of his eyes. The solution ignited the next step: formulation of GR, theory of gravitational fields, which was done again in exactly the same “vanilla” way. In Classical physics we have only two types of signals.

It is the whole foundation for Lorentz transformation... that a stationary body and a moving body should measure the same speed for light.

This is the principle of relativity (N. Copernicus) mentioned above that have nothing to do with the equations of motion; however it must be consistent with them. Think why we consider the Galileo law of inertia independent from the Newton second law (dynamics)?

The principle of relativity is the definition of the inertial frames (does not a matter Galileo group or Lorentz). And it is additional Principal Physical Postulate.

Our debate is pointless. In your post #26 you described exactly what was done by A.Einstein.

Regards, Dany.

P.S. Notice, that you start your threat with asking whether somebody will explain physics to you. And now you are in position to explain physics to me. There is nothing wrong with that. I am glad that it is so.

 

[SpitfireAce]

I was a tad frustrated, I'm sorry... I do appreciate your posts
I was trying to think of a good way to really explain special relativity to my dad... I am a firm believer that anything that is well understood can be expressed in a simple and very understandable way, and my dad doesn't know mathematics or physics (incidentally, either do I since I can't explain WHY it happens)...

when an observer clocks how long it takes a particle to go from a to b, the clock doesn't need to receive signals or information from the particle...
the particle and the clock run along independently of each other... the stationary observer connects them, he looks at the clock and looks at the particle and says that the hand on the clock moved this distance while the particle moved that distance between two simultaneous events (start and end of motion). But now for an observer moving towards the particle, the light carrying the information that the particle has started will reach him sooner than for a stationary observer, so the time that the hand on the clock started to move and the time that the particle started to move are no longer simultaneous (their finish times are no longer simultaneous either)... though they would look simultaneous since our perceptions don't take light travel time into account, thus under the assumption that the clock hand and particle started simultaneously (though they didn't), we see the effects of the unfair race as time dilation for the clock and length contraction for the particle.

*perhaps the time dilation in gravitational fields can be explained by a longer time delay caused by light having to travel through curved space-time to reach an observer in a gravitational field

I haven't thought this through completely (it seems that whether the obs. is moving towards the event or away should have an effect on whether time is dilated or contracted, but relative velocity between two objects moving apart is greater than rel. vel. if they are moving closer together so maybe the equations take this into account) but do you think its possible to describe time dilation and length contraction based on the principle of simultaneity (along the lines of reasoning above) or some similar logic or is it just something fundamental that one should just accept (like quantum wave collapse) that has nothing to do with the speed of the signal?

surely there must be some mechanism...

 

[Anonym]

Thanks for pointing out that C2 and C4 are the Pauli and Dirac algebras. I once knew that!

Sure, but I didn’t want to enter into discussion of the external degrees of freedom vs internal degrees of freedom (electric charge; E.Majorana particles are neutral) and the interconnection between them. We discuss here SR/CED/QED only and nature of time in it.

However, it remained not clear why I call the introduction of i=sqrt(-1) “stupid”. I do not know who did that. Perhaps, L.Euler. The matter is in it location on the time axis of history of science and the connection with the Trace operation.

The trace operation is crucially important in group theory and classical physics. Notice that the definition of trace as the sum of diagonal matrix elements is “philosophy”. It is not a math. The math operation must be defined uniquely. Now, consider for example A=2x2 matrix with the pure imaginary diagonal elements. Then in general Tr(A) not =0. But if you write the same matrix as 4x4 with real matrix elements, then Tr(A)=0.

To the best of my knowledge this mathematical mistake was corrected only by J.Dieudonne. Nevertheless it is still very common in current physical and mathematical literature.

Regards, Dany.

 

[mendocino]

SpitfireAce,

The reason why we subtract the time extension is that it works. In special relativity it turns out that quantity that is preserved in transformations of co-ordinates is the difference between the spatial extent and the temporal extent ( or interval). So in flat space

ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2

is invariant. Quantities that are invariant under transformations are important because they represent physical things rather than mathematical artefacts.

The reason why we subtract the time extension is that in special relativity, the speed of light is constant;
(dx^2 + dy^2 + dz^2) / dt^2 = c^2 , and ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 = 0 for light;

For all observers, it's a constant (zero) and is a special case of
ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 in general

 

[nrqed]

I was a tad frustrated, I'm sorry... I do appreciate your posts
I was trying to think of a good way to really explain special relativity to my dad... I am a firm believer that anything that is well understood can be expressed in a simple and very understandable way, and my dad doesn't know mathematics or physics (incidentally, either do I since I can't explain WHY it happens)...

when an observer clocks how long it takes a particle to go from a to b, the clock doesn't need to receive signals or information from the particle...
the particle and the clock run along independently of each other... the stationary observer connects them, he looks at the clock and looks at the particle and says that the hand on the clock moved this distance while the particle moved that distance between two simultaneous events (start and end of motion).

Sorry to butt in. I just hope I can help a bit.

I am not sure what situation you are considering here, exaclty. The clock is moving together with the particle (so the clock is in the frame of the particle)?

In any case, the start and end of the motion can *not* be simultaneous! In any frame. So I am not sure what you mean by "simultaneous".

But now for an observer moving towards the particle, the light carrying the information that the particle has started will reach him sooner than for a stationary observer, so the time that the hand on the clock started to move and the time that the particle started to move are no longer simultaneous (their finish times are no longer simultaneous either)... though they would look simultaneous since our perceptions don't take light travel time into account, thus under the assumption that the clock hand and particle started simultaneously (though they didn't), we see the effects of the unfair race as time dilation for the clock and length contraction for the particle.

Whenever an observation is made of a distant event (something happened over there and I have to wait to see the light from the event), there is some time delay, but the key point is that that time delay has nothing to do with all the strangeness of relativity. This is why it is important to introduce the notion of local observers. One must imagine an infinite number of observers located at every point in any given frame, with their clocks synchronized. One should really only talk about measurements made by local observers. This eliminates completely all effects of time delay due to the finite speed of propagation of light. When we talk about two events not being simultaneous in all frames, we do NOT talk about any effect due to the finite speed of light (which only make it seems AS IF two events are not simultaneous but it would not be a real effect).
[/quote]

*perhaps the time dilation in gravitational fields can be explained by a longer time delay caused by light having to travel through curved space-time to reach an observer in a gravitational field

I haven't thought this through completely (it seems that whether the obs. is moving towards the event or away should have an effect on whether time is dilated or contracted, but relative velocity between two objects moving apart is greater than rel. vel. if they are moving closer together so maybe the equations take this into account) but do you think its possible to describe time dilation and length contraction based on the principle of simultaneity (along the lines of reasoning above) or some similar logic or is it just something fundamental that one should just accept (like quantum wave collapse) that has nothing to do with the speed of the signal?

surely there must be some mechanism...[/QUOTE]
Again, you seem to think of time delays as being only due to finite speed of propagation. Even in special relativity, this is not what we mean by saying that time is different in different frames.

I think understanding the notion of local observers is a very important point in understanding what special relativity is really saying. I know that when I teach the subject, I put a lot of emphasis on this in order to avoid any misunderstanding about difference in time between two frames being simply due to the time it takes to *observe* something for a non-local observer.

 

[A.T.]

Apparently, when one calculates a length in flat space-time, one must add the lengths in the three spatial dimensions, and subtract... square root(-1)ct,

If you don't like the minus sign in:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2

you can rearrange it to:

dt^2 = ds^2 + dx^2 + dy^2 + dz^2

Then you see that ds, which gives you the proper time for timelike curves, is "just like a space dimension". If you use it as a dimension in a diagram, then the length of the curve is the coordinate time dt :
http://www.adamtoons.de/physics/relativity.swf

 

[country boy]

If you don't like the minus sign in:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2

you can rearrange it to:

dt^2 = ds^2 + dx^2 + dy^2 + dz^2

Then you see that ds, which gives you the proper time for timelike curves, is "just like a space dimension". If you use it as a dimension in a diagram, then the length of the curve is the coordinate time dt :
http://www.adamtoons.de/physics/relativity.swf

It should be further pointed out that the "interval" dt is not invariant. The invariance of ds as an interval is what makes the geometry possible.

Thinking of ds as a kind of space dimension can be instructive, however. One interesting question is: where is this spatial dimension when dx, dy and dz are zero, i.e., when the object is at rest? Is ds measuring a "hidden" dimension?

 

[Mnemosyne]

furthermore, why subtract? why is it that this weird equilibrium exists between space and time where motion through space takes away from motion through time (aging, time passage... you know what I mean)?
Why do we have this default time motion? The other dimensions don't work like this, it's not like I move up at full speed, but when I move right or left, I move up slower... the components are separate, like in projectile problems
on a somewhat separate note, why do photons have paths in space-time if they don't move through time?

I love your questions. :D

If time is to be treated as a dimension through which we (and all other known particles and waves) move (admittedly not a spatial dimension), the apparent slowing down of clocks at relativistic speeds is not an effect of that clock slowing down in time, but of its speeding up. Which leads to the problem of clocks not measuring motion through the time-dimension. The faster you go in space, the faster you go in time. (Although obviously not at the same rate. As can be seen from Ek = (gamma - 1) * mc^2 (relativistic kinetic energy) to double a particle's rate of "motion" through time, you need energy equal to the rest-mass (mc^2) of said particle. This energy would obviously give far more than a doubling in speed through space. :)

Photons do move in time. It's just that if they had wristwatches, they wouldn't seem (to us) to be working. They move very, very fast in time indeed.

Not sure what you mean by a default time motion.

(First post for me. Bracing for impact.)

-Mnemosyne

 

[A.T.]

Photons do move in time. It's just that if they had wristwatches, they wouldn't seem (to us) to be working.

It depends how you define "moving in time". For me a clock is "moving in time" if it is ticking. In that sense photons do not "move in time", only in space. But that's just one possible definition: Clock changes postion = movement in space; Clock runs = movment in time.

 

[rewebster]

What would we do if we didn't have clocks?