Exploring Imaginary Dimensions & Time Motion

[SpitfireAce]

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, my first question is...
is there some sort of physical significance to a dimension being imaginary or is this simply a mathematical trick so to speak?
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?

 

[Mentz114]

Probably the people who know about these things are not answering this because there are too many misconceptions in your questions. The time dimension is not usually taken to be imaginary in relativity.

There are too many 'why's as well. Actually no-one knows 'why' anything.

Get a straightforward book on relativity and don't ask 'why' too often.

 

[MeJennifer]

The time dimension can be taken as imaginary in special relativity (but not in general relativity).

Some professors prefer that due to practical advantages. See for instance Gerard 't Hooft in http://www.phys.uu.nl/~thooft/lectures/genrel.pdf

My stance is that in special relativity one should be comfortable with both approaches instead of praising one and dooming the other.

If you can use imaginary time in general relativity, where spacetime is curved, you deserve a price since no one has ever been successful in doing this.

 

[Mentz114]

SpitfireAce, you must be wondering if you can get a straight answer here without starting a squabble ( I refer to your other thread).

Let me try to give a more technical answer than I did last time.

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.

But as to 'why' his particular structure works - who knows ?

 

[Anonym]

Get a straightforward book on relativity and don't ask 'why' too often.

Get a straightforward book on relativity and you will obtain the perfectly rigorous physically and mathematically answer to your question: L.D. Landau and E.M. Lifsheetz, “Field Theory”, v.2; par. 1&2.

The definition of the interval is 1-1 consequence of the statement that there exists the upper bound for the velocity of the signal propagation (A. Einstein) which is now generally accepted as the Principal Physical Postulate.

i^2 =-1 is the notion introduced by stupid mathematicians (abstract algebra) since they were not able to understand that they should introduce the matrices and the matrix multiplication.

Sorry, but your post #4 is completely wrong.

Regards, Dany.

 

[Fra]

> Get a straightforward book on relativity and don't ask 'why' too often.

Or perhaps keep asking why is exactly they way to go, as perhaps you will actually start to understand WHY you keep asking why all the time. And that would be a substantial progress, and the result may be a refined version of the original why.

I was sometimes told as a kid to not "think too much", because it could make you go nuts.

/Fredrik

 

[Anonym]

I was sometimes told as a kid to not "think too much", because it could make you go nuts.

When I was a kid, I came to Professor V.N. Gribov and said that I want him to be my teacher. He said that I should pass the Landau-min exams (L.D. Landau and E.M. Lifsheetz, “Field Theory”, v.2 to begin with). I ask what his requirements are. He said: very simple, when you will finish study, you close it and write the same by yourself. Later he taught me to ask questions.

Regards, Dany.

 

[Fra]

Yes, that reminds me of a funny teacher I had in a relativistic QM course which was a professor. He seemingly didn't like to explain calculations to students that asked if they did it right, and he always seemed very bothered and responded that when you understand the topic you will no longer ask these questions, you will know wether it's right on your own.

/Fredrik

 

[Mentz114]

Dany:

Sorry, but your post #4 is completely wrong.

I don't see anything in your post that justifies that !

What is wrong with what I've said ?

The definition of the interval is 1-1 consequence of the statement that there exists the upper bound for the velocity of the signal propagation (A. Einstein) which is now generally accepted as the Principal Physical Postulate.

I bet you can't say why this so.

I'm not trying to discourage people asking questions - but 'why' is not the right question. If you mean 'why do we believe...' then that can be answered.

There have been other discussions on PF about this and the thing is to distinguish descriptions from reasons. Maxwells equations describe the EM phenomena exactly ( as far as experiments can tell) but tell us nothing about 'why' like charges repel.

M.

 

[Anonym]

What is wrong with what I've said ?

Sign.

Regards, Dany.

 

[Mentz114]

Dany: Do you object to the signature -+++ in the Minkowski metric ?

M

[edit] What I mean to say is that the interval is invariant whether one uses -+++ or
+---. It's a matter of convention.

 

[Anonym]

What I mean to say is that the interval is invariant whether one uses -+++ or +---. It's a matter of convention.

Not. Physics is the empirical science. The notion of the invariance (covariance) is defined in physics only with respect to the inertial systems which represent the observers. One can’t justify the existence of the observer the communication with him is impossible. The existence of the invariant quantities together with the infinite set of the independent observers allows defining the notion of the objective reality.

The best presentation of that I saw in the book referred above.

Regards, Dany.

 

[Mentz114]

I don't see the relevance of what you've posted to the sign convention of the metric. Nor does anything you say contradict any of my other statements about invariant quantities. Adds to it, maybe.

Let's call it a day - I don't know what we're arguing about.

M

 

[country boy]

There is a simple way to see why the time term has to be negative. If it were positive, then the speed of light would have to be imaginary. Just set ds=0 ...

 

[Feynman diagram]

When I learned relativity at university the Mathematics department taught us to use the -+++ sign for the metric in Minkowski spacetime, whereas the Physics department taught us to use +--- (on the basis that spacetime intervals would be >0 for timelike separated events, which seems a more logical way of thinking about it to me)

 

[MeJennifer]

For the theory it really does not matter what signature you use.

The best is to become comfortable with both signatures since they are both used in textbooks, papers and monologues.

 

[rewebster]

How can you write it (and make sense of how its presented) for both possibilites (and be variable for each/every sign) in the same/close amount of symbols? --or can it be?

 

[country boy]

i^2 =-1 is the notion introduced by stupid mathematicians (abstract algebra) since they were not able to understand that they should introduce the matrices and the matrix multiplication.

So Minkowski was a stupid mathematician?

 

[Anonym]

So Minkowski was a stupid mathematician?

You should not take my words too literally. I used it to emphasize the difference between the mathematics and the physics. The matrices were introduced by W.R. Hamilton (the physicist indeed). H. Minkowski obviously was familiar with them. Also no doubt the abstract algebra is very useful development.

My purpose was to point out that the physical requirements provide additional, very special restrictions imposed on the general mathematical consideration. For example, in my discussion above with Mentz114 clearly he is right mathematically (Noether “currents” and Casimir “charges”), but in the mathematics the notion of “observer” do not exist. That is the difference between the physics that deal with the “objective reality” and the mathematics that provide the infinite set of “subjective realities”.

The special relativity as well as the general relativity is the physical theories that use only the relevant pieces of the mathematical constructs. A posteriori you may claim correctly that the physical theory is one of the particular realizations among all possible mathematical schemes, but it is impossible to find it without physical consideration. In QM the observable is the self-ajoint operator and it eigenvalues must be real. Therefore, you may forget about the imaginary time. In addition, it demonstrates the intrinsic mutual consistency between SR, GR and QM.

Regards, Dany.

P.S. Notice that the length in physics must be positive definite quantity. Mentz114 post #4 is not the first that I enjoy reading. I am sure that he understood what I said; only he did not agree.

P.P.S. In order to appreciate how difficult is the problem of generalization, try to find the generalization of the matrix multiplication for the Cayley numbers.

 

[country boy]

You should not take my words too literally. I used it to emphasize the difference between the mathematics and the physics. The matrices were introduced by W.R. Hamilton (the physicist indeed). H. Minkowski obviously was familiar with them. Also no doubt the abstract algebra is very useful development.

Well, when Minkowski introduced i (as sqrt(-1)) he used the term "mystic." Not stupid, but maybe a bit spiritual.

 

[MikeL#]

ict & signature

0. I think there was an excuse for using i=sqrt(-1) between 1905 & 1915. It helped to show a ‘rotational’ symmetry between x & ict, it saved using a metric, and only a few mathematicians/physicists would understand the subject anyway.
1. But it introduced unnecessary abstraction or mystery (I agree with post #20). Even it you do use (x, ict), you have to translate it back to (x, ct) to apply to real measurements. And since GR, you need to use a metric anyway - so ict has saved nothing.
2. After advanced QM came in, I gather there is an excuse to bring back ict. But not in introductory treatments. (I am out of my depth here - but I believe there may be a connection between statistical mechanics, time, inverse temperature & sqrt(-1).)
3. Like post #15 I can really see why physicists like the +--- signature as it gives due emphasis on time.
4. But I sympathise with mathematicians who start with +++ 3d and add in time as an after-thought. The signs involved with momenta, Hamiltonians & Lagrangians are more friendly. For 5d and above, these dimensions have a spatial rather then temporal signature.
5. (I could be quite wrong here, but my own struggle in getting to grips with the Kaluza-Klein ‘miracle’ leads me to favour -++++ for 5d.)
Regards, Mike.

 

[country boy]

But it introduced unnecessary abstraction or mystery

I agree. Abstraction and mystery can get in the way.

Imposing mathematical formalism on physical theory is what makes it work. However, there is a tendency to regard the beauty in the mathematics as some kind of metaphysical truth, and this can mask the physical unknowns that remain. An example is the casting of relativity in terms of geometry, thereby eliminating any discussion of the ether. Although the early concepts of the ether were wrong, there does seem to be something there (gravity, the vaccum?). In solving physics problems we should strive to see past the mathematics.

 

[Anonym]

After advanced QM came in, I gather there is an excuse to bring back ict. But not in introductory treatments. (I am out of my depth here - but I believe there may be a connection between statistical mechanics, time, inverse temperature & sqrt(-1).)

No. It is not right place. It was clearly demonstrated by C.N Yang and R.L. Mills, Phys. Rev., 96, 191 (1954) that imaginary units are connected with the phases (internal symmetries and fundamental interactions).

Abstraction and mystery can get in the way.

Unfortunately, I have no idea what notion of “mystery” mean. On level of pure speculation I would say that it express the religion attitude which has much deeper roots in the human consciousness than the science.

Regards, Dany.

 

[Mentz114]

Can we agree that using i is always a mathematical trick and that any physical theory using i can be rewritten without i ( probably less elegantly) ?

 

[Anonym]

Can we agree that using i is always a mathematical trick and that any physical theory using i can be rewritten without i ( probably less elegantly) ?

I don’t know yet. My intuition says that the answer is no. E.C.G. Stueckelberg demonstrated that the answer in Quantum Physics is no, but I guess that is not what you are asking. It is directly connected with the nature of time in the Schrödinger equation. Also I use for gravitation ei, i=1,2,…,7 with ei^2=-1. It can’t be written in terms of matrices with real matrix elements.

Regards, Dany.

 

[SpitfireAce]

"SpitfireAce, you must be wondering if you can get a straight answer here without starting a squabble"
I don't think that happened mentz =)

thanks for the book reference... my copy is on its way

thank you all for your posts

Einstein derived SR from the nature of light (law of propagation)...

This was rather indirect logic since SR is fundamental and is not an effect of electromagnetic waves (SR is more than an optical illusion after all)

Is there perhaps a more direct way to theorize the connection between space and time and formulate SR independently of electromagnetism?
Perhaps such an approach, aside for being aesthetic, would give us greater understanding

 

[Mentz114]

Einstein derived SR from the nature of light (law of propagation)...

This was rather indirect logic since SR is fundamental and is not an effect of electromagnetic waves (SR is more than an optical illusion after all)

Is there perhaps a more direct way to theorize the connection between space and time and formulate SR independently of electromagnetism?
Perhaps such an approach, aside for being aesthetic, would give us greater understanding

It does seem strange that something as fundamental as SR should be deduced from light propagation, but I can't see how to separate them. Even in GR, light still plays a special part. The speed of light is related to two fundamental constants $$\epsilon_0, \mu_0$$, and these may be properties of the vacuum in which light propagates, so maybe the vacuum is more dundamental than either light or SR.

I don’t know yet. My intuition says that the answer is no. E.C.G. Stueckelberg demonstrated that the answer in Quantum Physics is no, but I guess that is not what you are asking. It is directly connected with the nature of time in the Schrödinger equation. Also I use for gravitation ei, i=1,2,…,7 with ei^2=-1. It can’t be written in terms of matrices with real matrix elements.

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.

 

[meopemuk]

Can we agree that using i is always a mathematical trick and that any physical theory using i can be rewritten without i ( probably less elegantly) ?

I think you are correct. The role of any theory is to describe/predict experiment. Experimental values of observables always belond to the real line (think about the dial of a volt-meter, or something of that sort). So, ideally, we should be able to do physics without complex numbers.

A common counterexample to this statement is quantum mechanics, which is formulated in the complex Hilbert space. 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. So, I wouldn't even try to calculate the spectrum of the hydrogen atom by using orthomodular lattices. Though, it is possible, in principle.

 

[Fra]

I read the original question again... here are some philosphical comments.

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)?

I don't know what you mean but nothing stops me from making a guess, even when I'm wrong :)

I can only speak for my own understanding and I like to think of the reason for the connection between time and space as beeing a consequence of the fact that any observer needs to inform himself of both. Moreover, any observer needs to inform himself about about reality itself, by interacting with it.

If one takes this view, the contraint of informing yourself has some implications if the view is to be consistent. When we read a clock, and compare it with the reading of your ruler, the situation is fairly symmetric from an information point of view. When we collect several pieces of information, this information has no real meaning on it's own, it's mere existenec is only related to the other pieces of information we have, partly store in our own state (memory). But before we get that first, we must be able to distinguish the ruler from the clock to a reasonable degree of certainty.

If one in the classical case, and assumes the time is a parametrization of the most probable disturbance given the initial conditions, then the change in the clock device is by construction, always <= probable than the direction of peak probability. If one then consider a "surface" of equiprobable disturbance in information, and adds to that that a large change in any state is expected to be less likely than a small change, then the result of the probability games seems to be an upper bound of the changes of unknowns relative time.

In the general case (QG domains etc) I think the upper bound is replaced by an expectation of the upper bound. The cause, and support of the expectation is the current information. I ultimately picture that this is also the way for dimensions to collapse, since when expectations are not well supported, the formalism for it, becomes destabilized, and down with it goes the notion of space.

I am well aware of that this is fuzzy, but this is a simple intuitive attempt to describe it. I am still working on a consistent formalism. But I am close to convinced it has to be possible.

In a way it's simple, but the reason why I'm struggling with it is because consistency suggest everything is related and I have a hard time even detaching the formalism for the subject of study. But this is also the exploit to connect spacetime with the objects there in, in a relational manner than connects to information concepts. So in the end it's not that simple after all, although the idea is intuitive.

I except this to yield a better understanding on energy/mass, entropy and time and typical attributes we usually assign to reality, like geometry, dimensionality etc and how they all might interact.

/Fredrik

 

[Anonym]

Einstein derived SR from the nature of light (law of propagation)...

This was rather indirect logic since SR is fundamental and is not an effect of electromagnetic waves (SR is more than an optical illusion after all)

Is there perhaps a more direct way to theorize the connection between space and time and formulate SR independently of electromagnetism?
Perhaps such an approach, aside for being aesthetic, would give us greater understanding

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.

By the way, notice that in my post #12 I use the notion of the objective reality suitable for the discussions of the extended objects (QM and fields in general). That definition is the natural extension of A. Einstein original communication requirements and rejects completely his later definitions based on erroneous interpretation of D. Hilbert spectral decomposition theorem.

It allows understanding the peaceful coexistence of the collapse of wave packet (point particle) with the experimentally measured picture of the single electron (wave) in double-slit set-up. No doubt that the result presented by A. Tonomura is reproducible (invariant) in any laboratory (inertial system) which is connected with his Tokyo facility through time-like interval.

Regards, Dany.

 

[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.