Schrödinger found his equation heuristically from the Hamilton-Jacobi equation, and with the idea of De Broglie.  He searched to fill in the details in a wave equation, whose condition of finiteness and single-valuedness  would give the quantization condition. He was full aware of the non relativistic approximation, and that the first order in time wasn't the last word.  All that is spelled out in his series of papers in Annalen der Physik (in German.)

Feynman points out the contrast between the Schrödinger equation that is a breakthrough in physics, and his own formulation in term of the paths integral that has no originality whatsoever, and is a mere mathematical formulation of what is already known about waves.

In analogy to a physical wave it is natural to represent it by an amplitude and a phase so in two dimensions. Complex numbers operate in two dimensions so they are convenient to use them in this case. I hope that this is explained more in detail in the basic quantum physics reference books.

The wavefunction is complex and  measurable quantities must be obtained from a probability distribution involving integrals of the product of the wavefunction and its complex conjugate. Whether or not you can think of the wavefunction as 'an element of physical reality' has been a long-standing source of conflict in attempts to reconcile the mathematics of QM with philosophical interpretations. In a sense, one can think of it as a diffusion of probability in imaginary time (and I think this can be seen from casting the Fokker-Planck equation in terms of an imaginary time?) There is a tremendous body of literature on the subject, most of it outside the scope of introductory QM coursework to be sure. Feynman had a pragmatic view point (paraphrasing him), that physicists can all do QM, but no one really understands it. 

From my view, mathematics provides tools and methods for physicists to describe processes in nature and to predict them. Complex numbers are just one, but perhaps one of the most useful tools here. If you consider the techniques of complex analysis, which have been very prudently used by Feynman and others e.g. throughout the whole QED, you will see that e.g. the integration of functions becomes a rather straightforward task considering the complex space. And this is an addtional reason to use complex numbers as the mathematics becomes partially simpler.

His equation should be in first order in time because it is non relativistic, since in this approximation the energy is linear.  The only way to specify completely the initial value is through two variables, and the most natural way is to use a complex function.  The Klein-Gordon equation is relativistic, and thus can be real.

The most commom way of representing quantum wave functions (and wave packets) is plane waves, which can most easily expressed matghematically by the exponential function. Only an exponential funciton with complex argument oscilates (i.e, can representa a wave function). Exponential functions with real arguments do not escillate - they either blow up or goes to zero for the argument going to infinity.

Dear all 

Thank you so much for your nice answers. I want to ask specifically about the time and position, can we consider the complexity of the wave function as an indication of the complexity of the relation between the positions and time of the quantum particles and the classical positions and time? 

For fuller answers to your original question:  See the Nobellist, C. N. Yang's, chapter "Square root of minus one, complex phases and Erwin Schrodinger" pp. 53-64, for an historical and low-level technical view, in the book "Schrodinger Centenary celebration of a polymath" ed C.W. Kilmister, Cambs U Press 1987, reprinted 2008.                       For a more detailed technical explanation, in terms of the relative phases and the wave function, see Chapter 10, "Why is the State Complex", in "Essential Quantum Mechanics", pp.128-144, by Gary E. Bowman, OUP 2008.  

Enjoy!

Update 2 Oct '14: Attached Yang's chapter referred to above - see especially pp. 54 - 57

Further update 6 Oct 2014: See notes from Oxford U "One day seminar on Complex Numbers in QM" especially the more formal list of abstracts of speakers and references on the final page. But be warned, it's quite mathematical. See especially p. 13 of the Oxford seminar paper as regards the discussion (here) of whether or not one can satisfactorily decompose complex equations into pairs of real equations.

http://www.damtp.cam.ac.uk/research/gr/members/gibbons/Oxford.pdf

Yes, we can Sadeem. Relativity is also simpler, if time is imaginary.

Regards,

Eugene.

@ Claude Pierre Massé , Nevertheless the Dirac equations are relativistic but are both linear in time and include imaginary term, so your conclusion does not seem logical. Psi in the Schrodinger equation denotes a " probability " amplitude and not a real wave amplitude , otherwise it coincides the " diffusion " or " wave propagation" equations and could not be interpreted as probability amplitude in Hilbert space.

The Dirac equation is the square rooth of the Klein Gordon equation, of second order is time and space. It has 4 component because of its first order, that's more complicated.  The electromagnetic field is real, yet a probability density can be defined from it, and that's true for a mechanical wave too.  In those cases, the time derivative is used.  That's the complete set of misconceptions.  Quantum mechanics as wave mechanics is no different from classical waves.  The difference is in the phase velocity and the projection postulate, otherwise it is pure linear algebra and calculus.  It is but a useful mathematical trick to transform a second order differential equation into two first order ones with an additional variable.  But in the case of the Schrödinger equation, it is already of first order because of the non relativistic approximation.

Thanks to Paul G. Ellis for the references.

Eugene, and others

I have found a complex form of Lorentz transformations and I used it to interpret the remote interaction of entangled particles. The complex transformations supports what you are saying. It can be a good interpretation as a reason of complexity of the wave function, what's your opinion?

the link is:

https://www.researchgate.net/publication/259996539_Solving_the_instantaneous_response_paradox_of_entangled_particles_using_the_time_of_events_theory?ev=prf_pub

Article Solving the instantaneous response paradox of entangled part...

There is a problem with many "times", Sadeem. Time is connected with existence and  many existences are forbidden (by sense and definition of existence). In any case your paper is interesting.

Regards,

Eugene.

I believe that the origins of the Schrödinger equation came up briefly in my undergraduate and graduate physics education. While the consequences are profound, as several of the responses that you have already received indicate, the origins as I recall their being explained to me were quite simple. Schrödinger made an inspired guess.

I will attempt to recreate that.

The classical wave equation is second order in its time derivative and second order in its spatial derivative. Solutions to the classical wave equation are real-valued functions (amplitudes - the physical displacement of a element of water in a pond, for example, from its rest position) of space and time. These facts allow a second time derivative of the solution and / or a second spatial derivative of the solution BOTH to give you a function of space and time that is also the original solution, to within a coefficient. These two means by two second derivatives to get the wave solution "back" (within a coefficient) is what has the classical approach to waves "work", mathematically speaking.

I hope that I am remembering this properly!

Schrödinger was also working in the dimensions of space and time. However he was not working with physical displacements in terms of the solutions, or wave functions, to his, at that point, unidentified wave equation. He was (or eventually would be) dealing with a wave function whose dimensions were the square root of probability density. Yet, he still wanted to reproduce the behavior analogous to the classical wave equation. The best he could come up with however was an equation that was FIRST order in its time derivative and, like the classical wave equation, SECOND order in its spatial derivative.

I may have forgotten a piece here. I cannot seem to recall why he needed the time derivative operator to be first order. Perhaps someone can fill in this missing piece.

So, he needed a wave function that he could get "back" (to within a coefficient) after only ONE time derivative and, as before, after TWO spatial derivatives (to within a now complex coefficient). One way to do this was to introduce i (the square root of -1) into the argument of his trial wave function and in front of the time derivative operator in his wave equation. Then one could get the solution "back" after only ONE time derivative and, as before, after TWO spatial derivatives.

The only price I am aware of that he paid for this was the fact that the wave function was now no longer an explicitly real-valued function and his equation had an "i' in front of its time derivative operator.

At a practical level, one can, if one chooses, ignore any number of deep physical and philosophical issues by saying that when one calculates expectation values using this formalism, one calculates intensities, not amplitudes, that is, one usually writes integrals containing the wave function and its complex conjugate that are "squared" in that sense, and one ALWAYS obtains a real answer. So, that little question of what does it mean to have a wave function with an "i" in it lies conveniently in the heart of the calculation but never in the results of that calculation.

Since the calculations agree well with the measurements, one can get away with ignoring this problem, or one can study and contemplate it. It is a personal choice. For some reason, the majority of working physicists are quite comfortable to be able to do the calculations, while an interested and dedicated minority continue to push back on the boundaries of what the "i" means. They advance physics in so doing.

I will end by skipping a great many steps and point you to a program that picks up on a related conversation almost a century after it began with Schrödinger and others: http://www.youtube.com/watch?v=GdqC2bVLesQ

Thank you for a wonderful question. I hope that my recollections are correct. (Feel free to check them. This is oral history. Who knows what I have got wrong or glossed over.) I also hope that they are useful.

Dear Sadeem, As regards your question about Schrodinger 's equation and the remarks  by  Dr. Masse  about Dirac's equation, may I suggest that you write to my friend Prof. Hossein Javadi , see http://cph-theory.persiangig.com. He is an expert in Relativity theory. Quantum Mechanics,etc and you may mention if you like that I suggested that you should contact him. I am having   health problems and  I retired 12 yrs ago and so  I am unable to answer the questions being asked.  Wish you all the best in your quest for foundations of  Quantum Mechanics and Relativity. Best regards.

Prof.Dr.Gulzari Malli, PhD 1964 ( U of Chicago)

Dear Eugene

"time is connected with existence and many existences are forbidden"

But, at the same instant time is connected with speed and movement. So even if objects has different times due to their differences in movements they all can exist together at the same space. in fact the two times are conjectured with spatial dimensions splitting, that makes us to observe the projection of quantum particles instead of observing their real movement direction. Because they has more dimensions that we can't recognize in our four dimensions space time.The mechanism of movement is as follows:

When a photon is moving in its direction, we see this movement as a moving in space. While actually its precede us in time and what we see actually is only the projection of its movement. This what makes the transformations complex between the quantum particles coordinates and our coordinates. At last its one option that may explain the cause of complexity of wave function and the causes of uncertainty in positions and time of these particles in a non conventional way.

regards 

Thanks Mark for your detailed explanation.

Dear Professor Gulzari L Malli

Thank you so much. I wish you full health and full recovery. Thank you for your interest in my question, I will send a message to Prof Javadi

I appreciate your participation.

Thanks

Dear Sadeem, If you hear from Prof. Javadi   about  the question being discussed, you may ask Prof. Javadi  the e-mail  of my very closed friend ( for ~ 60 yrs!)  Prof. Pran Nath, who is Marshall Professor of Physics , Northeastern University, Boston, USA.  He is the originator of SUSI ,etc and written up in  Wikipedia. You can search him  on Google (search  as  Prof. Pran Nath, Physicist at Northeastern University) and you will get an idea of the width and breadth of   Pran Nath 's knowledge  of fundamental problems in  "Physics".  If Prof. Javadi has not heard from Prof. Pran Nath ,then either he is not interested ( I doubt it very much) or he is too occupied in his own research  ( he is 75 yrs young!).  I do not know personally any one better than Pran Nath  amongst the living  physicists who I think can answer your questions.  Prof. Gregory Breit ( while he was alive)  was one of the most prominent physicists I have had the privilege of working with ( in atomic physics) in 1964-1965 as a Research Associate.  I am sure Breit would have loved to hear ( and most probably)  answer  your questions. I  am really pleased to learn that you are asking very fundamental questions and I hope very much  that solutions   would be found  during the dialogue of this Forum.  I would ,however quote Lev Landau  ( doing from memory) " The brevity of my life does not afford me the luxury of worrying about problems for which I can  ( or will?) not find solutions in my lifetime". Wishing you all a very long life so that you may find solutions to many questions(problems?)".

 With best wishes to all the participants in this discussion.

  Prof. Dr. Gulzari L. Malli 

Hamiltonian, H, is the operator for the infinitesimal translation in time; given a state at time t0, that is given |Ψ(t0)>, the state at time t is obtained from the expression  |Ψ(t)> = exp(-i H (t-t0)/ɧ) |Ψ(t0)>, where ɧ = h-bar. The imaginary unit i in the exponent is required in order for the time-translation operator exp(-i H (t-t0)/ɧ) to be unitary. The Schrödinger equation is nothing but writing the expression |Ψ(t)> = exp(-i H (t-t0)/ɧ) |Ψ(t0)> in differential form. Conversely, the latter equation is the solution of the time-dependent Schrödinger equation subject to the condition that at time t0 the state of the system described by H is |Ψ(t0)>.

Dear Sadeem,

It seems we have all forgotten the real solutions to Schrodinger equation e.g. in the case of particle in a box or in  the Harmonic oscillator case or the H-atom even. Although wave functions in general are complex there are situations galore where we have real wave functions and if you are bothered about harmonic time dependence, it can very well be represented by sine wt or cos wt depending on the phase.

Complex numbers of course make calculations convenient.

Regards,

Rajat

Dear   Prof. Dr. Gulzari L. Malli

I contacted him and waiting for response, thank you so much.

wish you all the best.

regards

Sadeem

Thank you Behnam Farid

Dear Rajat Pradhan

Yes there is special cases of course, but the general one is the complex.

Rajat Pradhan in above : simplification of calculations. YES. This is it. Signals are transferred from Time domain to Phasor domain in which the sinusoidal signals are represented by a rotating phasor in the complex plane with magnitude corresponding to the signal amplitude and angle representing the phase shift with respect to some reference signal at angular speed ω rad/sec (counter clock wise). At any instant, the real signal is the projection of the rotating phasor on the real axis ( A.cos (ωt+φ)= Re( Αe^(jωt+φ) ). Of Course, if we drop the ω (the angular frequency) assuming all signals have the same frequency results in that all signals can be represented as tensors with amplitude and phase angle on the complex plan (like vectors). Hence, the complex continuously changing time domain signals can be treated as numbers (but complex numbers) which simplifies calculations or as vectors (which leads to graphical addition,...etc). Thanks. @Aldmour.

You are welcome Sadeem.

Scrodinger's equation can be  written as   TWO   "real" non-linear equation (two, conservation of   mass and momentum)  [ see Dirac's Text Book].  Scrodinger's description is simpler, being  linear; requirement of "i"  is  only convenient !

POINT IS :   To describe anything  you need language,  use a "complex" one to keep the description simple  or  simple (real) one that makes  the description  complex.

It is like,  making a  choice :  "2488", OR  "MMCDLXXXVIII"  or " two thousand four  hundred eighty eight."

Look at an interesting discussion at :

https://www.researchgate.net/post/Why_do_you_need_imaginary_numbers_to_describe_Quantum_Mechanics2?_tpcectx=profile_highlights

I find it instructive to see how many different directions people are taking to approach this question of complex numbers and their use in QM.

Another direction, not explicitly mentioned so far (I think), relates to the description of waves such that they incorporate position-momentum uncertainly (Dicke & Wittke, Intro to QM, Addison Wesley, 1960 and ff., p. 24 in the 1969 Third Printing – and showing my age):

“…as will be discussed in detail later, there are reasons for believing that a particle with its momentum exactly known is in a state such that its position is completely uncertain. In such a case, the probability distribution measured by the wave intensity | ψ |^2 should be independent of position. This suggests that the wave function to be associated with a particle of definite momentum should be given by Eqn 2-2: ψ = A exp[ i (kx - wt)], rather than by Eqn 2-5: ψ = A sin (kx – wt + α)].”

As already mentioned, all observations record real numbers. So Is it, in the end, simply a matter of taste as to whether one uses one complex variable to encode a pair of variables in one equation, or uses two sets of equations?

Personally, I like to view the complex i (and its various hypercomplex manifestations) as something allowing an easy encoding of direction, phase, rotation etc., as distinct from magnitude.

(Posted initially, but unintentionally, at PK Mohanty's "interesting discussion", linked in the post immediately above this one (but it seems relevant to both)

Paul,

i and hypercomplex are real symmetries of (mathematical?) world. We can not ignore them.

Regards,

Eugene.

I think it is an interesting question. We can widen the scope of the question a little by asking if we can describe nature (at quantum level) without using complex numbers, or stated differently: Does quantum theory need complex numbers? I think the answer is yes. The situation here is essentially different from electrical engineering where complex numbers (phasors) are a convenient tool to make calculations easier. However, in quantum mechanics, the complex numbers arise at a more fundamental level: For example,  Feynman used as a the starting axiom  that each path in the path integral contributes a weight exp (iS/hbar) . It is difficult to replace this axiom without reference to complex numbers. Another argument in favour of using complex numbers is that Heisenberg's matrix mechanics implies that observables are associated to linear operators. These linear operators act on complex spaces and not real ones. The reason is that the translation operators in space /time are of the form exp(iP.x) and exp(iHt). Again the complex number pops up. By the way: the fact that quantum mechanics really needs complex numbers also means that we have never understood quantum mechanics at such a comfortable level as for example relativity theory.

Sadeem,

your ideas (many times) go far beyond traditional QM and relativity. To be sure, that they are correct, You must test them on well known problems, such as hydrogen atom and so on.

Regards,

Eugene.

Dear Eugene

These ideas are fully consistent with quantum mechanics I can derive the complex form of wave function using these transformations, with out need to add hoc the complex i to these transformations and they actually provide the answer of the question; Why the wave function is complex?

Applying it to hydrogen atom will not bring any thing new because they are relativistic transformations and they should be approximated to the usual solution in this case. Nevertheless I will take your advise into consideration in applying it to high energy applications.

Thanks a lot.

regards

Sadeem,

and 2s-1p splitting?

This question is more profound than it first appears. It reflects the unease everyone has suffered at the thought that reality is represented by imaginary numbers. Shut up and calculate never quite releases us from this unease.

The scalar square root of -1, i, is purely imaginary, a mathematical fiction. However, there are real square roots of -1, for example the unit bivector which contains geometrical information. The geometrical algebra olf a plane defined by unit vectors, a and b has four elements; the set of scalars, the vectors, a, b and the pseudoscalar i = a  ^ b. This is why i turned out to so useful. It contains all the geometrical information about the geometry of a plane and represents orientated rotation in the plane. The complex conjugate -i is rotation with the oposite orientation.

When we take a measurement of a quantum system, we take the scalar product of the wave function and its complex conjugate and performing the product < i | -i > = 1 thus rejecting all the geometry of a plane in favour of a scalar number. The proponents of quantum discord call this lost information discord.

The result of replacing imaginary i with the unit bivector is to reproduce the EPR experiments with a hidden variable, the orientation of the bivector. Joy Christian (check him on arXiv) did this thus removing the spooky action at a distance and made nonsense of the ontological speculations derived from the Bell-Kocken-Specker theorem.

Ask not why the  i, rather ask which multivector quantity it represents and what information we are rejecting in assuming the imaginary scalar i.

The first chapter of Sakurai's QM text "Modern Quantum Mechanics" is helpful to understand this question.  If we assume that the states of a physical system are represented by vectors in a vector space, the Stern-Gerlach experiment shows that the states of a spin-1/2 particle are elements of a 2-dimensional vector space.  There are three components of the spin angular momentum, each of which must have its own basis states.  If one tries to restrict the vector space to have only real-valued scalars, there are not enough basis states to describe the relations among all three components of angular momentum.  With complex-valued scalars that can be done.  

Eugene, Andy, WilliamK,

Though not myself an accomplished mathematician, I’ve long been interested in the meanings and necessity, or otherwise, of complex i in the description of physical systems, whether as simple complex numbers, or as multivectors etc.

Initially, I thought the simple complex number i unavoidably unique, partly for the same reasons as the OP, and also as given by e.g. Dicke&Wittke and others. But after learning about the wide variety of ways available for encoding geometrical information - not merely vectors and tensors, but also spinors, paravectors, and the whole panoply of Grassmann/Clifford/geometric algebras, it seems that there are a great many ways to encode (and I use that particular word very deliberately) the various geometrical properties, as the application requires (e.g. paravectors are well suited for special relativistic applications - see e.g. W. E. Baylis).

However, and in reply to one of Eugene’s earlier posts (3rd above, counting up from here), I agree completely that we cannot ignore the real symmetries embodied in hypercomplex numbers. But in all honesty, I find myself wondering whether hypercomplex numbers are ‘simply’ hugely-more convenient encodings than working with sets of reals that, ultimately, might also be capable of modelling the same symmetries. My conjecture is that reals could do so, but would be impractically unwieldy for situations of much interest.

WilliamK, thank you for suggesting what might be a suitable starting point – to look for a way to encode spin information in a 'real' space/structure of some sort. (Just don't hold your breath waiting for me.)

http://arxiv.org/abs/1211.0784

Dear Eugene 

its hard to find any difference between these transformations and the conventional one in atoms spectra, because the effect of Lab movement is usually neglected. But The effect can be more apparent in high energy experiments.

Actually complex number is a pair of real numbers, which commute in a certain way. So a complex number has two different and independent dimensions, one is magnitude and other is the phase. In case of dual nature of matter, particles are define as wave propagation. The wave propagation is extremely well defined in terms of complex numbers as an bounded cyclic magnitude and a space-time varying phase. It important to note that two different waves might have same magnitude but different phases and a super-position of them depends not only on magnitude but also on the phase. Remember "destructive and constructive interference"?

We can analyze the wave propagation even in terms of sine and cos and vectors but complex numbers as an exponent of imaginary number are much more canonical and easily to handle representation.

What the above comments say is correct: having a complex function is the same as having two real functions. Which two are appropriate? The real and imaginary part surely are not, since the psi function is defined up to a phase factor, which mixes the two. Norm and x-dependent phase is much better, but these satisfy nonlinear equations.

But why do we need two functions in the first place? To this the answer is surprisingly simple. Quantum mechanics needs, after all, to become equivalent to classical mechanics in an appropriate limit. But classical mechanics requires both information on position and *velocity*. A single psi function would only give information on one of the two. Thus the norm of the psi function only tells one, where the particle will be (in probability) but not the velocity it will have. The space dependent phase contains some amount of velocity information. It turns out that these two kinds of information can be ncoded in one single complex function.

Best wishes, Sadeem. I'l be glad, if You manage to incorporate your views in modern physics. The main it's problem is overcomplexity. If your views will simplify it, then it wiil be good, if not, then not.

Paul,

from mathematical point of view i and hypercomplex are peculiarities of low dimensionality. Low dimensional spaces in some sense are degenerative and possess special symmetries, i in 2 dimensions, quaternions in 4, octaves in 8. Higher dimensions have no these symmetries. Physics only uses this fact for it's purposes.

Regards,

Eugene.

Eugene

As you indicate, the division algebras allow richer symmetries than those to be found in higher-dimensional structures.

But what happens in some of the higher-dimensional theories of physics? In my rather limited understanding, they rely on Bott periodicities, in effect, as I see it, taking at least some of the richer symmetries into higher-dimensional structures? (Some refs linked below.)

Regards – Paul

http://ncatlab.org/nlab/show/division+algebra+and+supersymmetry

Dear All,

The Schrodinger prescription works only for free particle and scattering states which are eigen states of momentum and energy both and which allow complex wave function representations. For bound states like those for particle in a well, H-atom, or harmonic oscillator, we don't have definite momentum eigenstates which are simultaneously eigenstates of energy. 

So it is not a very general observation to say that Schrodinger put the "i" on purpose and that otherwise things would not have worked out. Even for those ones where the i is inserted we can do QM perfectly well with all real wave functions sine or cosine in place of exponentials.

Regards,

Rajat

Yes, we can, Rajat, but for what? Let us try to make life and science simpler, not harder.

Yes Yes  Eugene. That is right, of course. Simpler life with complex numbers, and complex life with simple Reals.

Regards,

Rajat

Rajat: what you suggest may sound simpler, but in fact it is wrong: the probability of finding a particle at position q at time t is given by the squared modulus of psi(q, t). For a stationary state, psi(q, t)=psi0(q) exp(i omega t), so the probability is time independent. If you were to multiply by cos(omega t) instead, the total probability would not be conserved and actually vanish at some time. It is not impossible to make a real QM, but you must replace every wave function by a pair of wave functions, so the gain is rather illusory. In any case, it is essential to have some rule stating what the probabilities are, which will usually involve the wave function intensity, that is, some form of squared wave function. It is then important that the time evolution should preserve the full probability, since it is clearly necessary that the probability of finding the particle somewhere shoulsd always be one. The way in which these assumptions lead to a complex Hilbert space, a unitary evolution and thus a Schroedinger equation that involves an i is maybe not totally convincing, but it is surely the simplest way to solve the problem.

Another reason why we need an i is in order to encode the velocities as well ast he positions, as I pointed out in an earlier post.

Let us differentiate between entities that are a kind of macroscopic entities from the ones that are particle-like. We shall indicate that complex numbers are a micro-world assortment of the reals!

In the following picture from my book ΘΕΜΑΤΑ ΣΤΙΣ ΘΕΜΕΛΙΩΔΕΙΣ ΕΝΝΟΙΕΣ ΚΑΙ ΤΑ ΘΕΜΕΛΙΑ ΤΩΝ ΜΑΘΗΜΑΤΙΚΩΝ, 

https://www.researchgate.net/publication/243962466_________  p.216, Section 8.3, The true nature of complex numbers, δακτύλιος means ring,  υπερφίλτρο ~ hyperfilter, 

μεγιστικό ιδεώδες~ maximal ideal, σώμα~field, φανταστικός άξονας~ imaginary axis.

Compairing the two constructions, since infinitesimals are in a "microscopic level of reality, we conclude that complex numbers are in a kind of microscopic lever of the reals! 

The final conclusion is obvious: Particles or waves are also in a microscopic level of reality, like the complex numbers, and so it is natural to perform operations.

Book ΘΕΜΑΤΑ ΣΤΙΣ ΘΕΜΕΛΙΩΔΕΙΣ ΕΝΝΟΙΕΣ ΚΑΙ ΤΑ ΘΕΜΕΛΙΑ ΤΩΝ ΜΑΘΗΜΑΤΙΚΩΝ

Costas,

complex numbers is a special symmetry of abstract two-dimensional linear vector space (pure algebraic feature). Sizes are no matter here. See Paul.

Regards,

Eugene.

Already classical mechanics can be formulated as a function of pairs of angle and action variables.  It has been claimed that the cosine hasn't a constant amplitude, but let's consider a classical wave with extension y.  The energy density is the sum of the potential energy k/2 y^2 and of the kinetic energy m/2 y'^2.  If y is a cosine function, y' is a sine function and the energy density is a constant for a plane wave.  The probability density is proportional to the energy density, tha's basically the same thing.  The complex numbers aren't consubstantial to quantum mechanics, but to differential equations.  To have existence and evolution, there must be a principle of change and a detail of what changes (Leibniz)

I agree with you Arno

I think the complexity of wave function and quantum mechanics is an indication of a deeper philosophy of a complex transformations between the frames of quantum particles and the classical objects' frames. The mathematics of differential equations are just a result of this complex relation. 

The principle of change is broadly forces, and the detail of what changes is broadly motion.  Forces make potential energy "disappear" into kinetic energy, and conversely, motion makes kinetic energy "disappear" into potential energy.  Conservation of energy is a convenient way to describe existence and evolution.

Dear Eugene,

It would be profitable for you, if you try to understand what I said! You statement

"complex numbers is a special symmetry of abstract two-dimensional linear vector space" does not really give any deeper understanding of complex numbers. I can define complex numbers as 2x2 matrices:

Dear Charles, 

My intension was not to make things more difficult. Simply I interpret complex numbers, as a kind of "particles" inside real numbers. The imaginary axis goes down to microscopic levels of reals. I suppose that this is a new approach to complex numbers, very convenient for teaching as well.  

They are not only ordered pairs, but they is an addition and a multiplication law.  Why add a complex multiplication law while there is the one of matrices already?

@Arno: You say: ``Waves are mathematically described with a real and an imaginary portion. Physicists have decided arbitrarily to disregard the imaginary part. This part is imho necessary for understanding non-local phenomena though.''.

In fact, if one has an ordinary wave, which describes a quantity such as velocity or pressure, described by a real number, then for each solution of the wave equation, the complex conjugate is also solution, and hence the sum of the solution with its complex conjugate, as well as (i * the difference) are solutions, by linearity. There is thus no issue of dropping the imaginary part: both real and imaginary part are legitimate solutions. None are discardds. it is just sometimes easier to get the complex solution first and then take either real or imaginary part.

On the other hand, for the time dependent Schroedinger equation (the only physically significant one) it turns out that the equation itself is complex, that is, if a solution of the Schroedinger equation is found, its complex conjugate is a solution of the time reversed equation, not of the equation itself. Again, this relates to the fact that spatially dependent phases are related to velocity. This is the basic reason why we need complex numbers. This is in particular also the case for single particle Schroeinger equation, for which nonlocality is not really an issue. 

Dear F. Leyvraz 

you said "this relates to the fact that spatially dependent phases are related to velocity. This is the basic reason why we need complex numbers. "

That's what the mentioned transformations refer to. Some of the researchers asked me, why they are complex transformations? why don't you just use the real one?

The problem is when the real one is used like the conventional Lorentz transformations for quantum particles relative to classical observers, then there will be zero uncertainty for time and position, which is not allowed in quantum mechanics. So, there is no escape from using the complex transformations between quantum particles and classical observers. By the way, these transformations are converted to conventional Lorentz transformations if they are used between two classical observers.

Charles ;

I'm not talking about waves, I'm talking about the relation between time and position of quantum particles and classical observers. The Hilbert space is only a mathematical space not a real space and the probability is only a sign of a deeper physics which is a complex relation between position and time of quantum particles and our time and position ones.

Yes, You can, Costas, but You must postulate their properties wich is equivalent to mentioned by me symmetry. Ask You once again to see Paul G. Ellis.

Regards,

Eugene.

Charles;

"but it enables calculations on an underlying reality in which we construct the notion of time and position in our world, but in which these concepts do not apply to quantum particles. This is not a complex relationship, but an underlying reality so simple that such a relationship does not exist."

This is not make sense. How time and position do not apply to quantum particles? What about uncertainty principal which involves time and position?

It's better to say it's applies but in different way and manner that leads to make their relation a complex one with respect to our ones.

This makes sense, Sadeem, on philosophical level.

Regards,

Eugene.

Arno,

nothing does not exist by definitions of the terms "nothing" and "exist. It is impossible logicaly and, as You say above, does not exist.

Regards,

Eugene.

Hi Arno Gorgels, how do describe logical description? I think all descriptions which do not contradict observations and are logically not inconsistent within itself are logical descriptions. In fact look at my this paper: https://www.academia.edu/4950937/Converting_Liar_Paradox_to_Incongruent_Set. In this I conjecture that the existence of liar paradox and in-consistent/in-congruent statements is the source of all knowledge and meaning in the universe. If everything was in agreement, there is nothing to discover, no knowledge.

Either the matrix multiplication, or i^2 = -1 must be introduced.  The former is obviously simpler and more natural than the latter.  Only after comes the abstraction, i is just an alias for a special 2x2 matrix, and everything is found again just by linearity.  Matrix algebra has much more applications than the complex numbers and is the natural language of linearity.

Charles; and others

"Sadeem, In qm we have the properties of time and position only in measurements. We do not have them between measurements. This is the root of the uncertainty principle which applies to measurements."

This is the Copenhagen view, which wants any one to deal with quantum mechanics as a non logical science. The particle is there if you measure it or not and as soon as it's there it must be governed by its own time and position. This is part of the philosophy of "Sharp and calculate" this phrase unfortunately freeze our minds and make us deal with quantum mechanics without any logic. This phrase was justified at the beginning of quantum mechanics appearance, but it can't be applicable forever . There must be a certain physics that can connect between our time and position and the quantum particle's time and position. These transformation fully agree with uncertainty principle.

Quantum mechanics can't work without an evolution equation that predicts the probability of every outcome, including the measurement of position and time.  But that is logically inconsistent with the inexistence of the particle, including space and time, between measurements, because this equation must be expressed in space and time.  In the formalism, the axiom of unitary evolution and the one of projection are contradictory, as nothing determines which one must be used.  This fundamental flaw has been concealed by various methods, like "shut-up and calculate," positivism, complementarity etc. because indeed, when the ad hoc axiom is used, verified predictions are made.  But it is an illusion, in a really new situation, no prediction is possible.  When the good choice is finally done, it is claimed quantum mechanics is a fair success and the sceptics are wrong.

Dear Charles;

There is something that is governing the whole universe which is time. Every particle is governed by time from the photon to the super massive black hole. The problem we just think about time as simple as a recording of our clocks. It's not as simple as that time is related to dynamics, every moving particle is affected by time. We can't understand time unless we understand the dynamics of these particles. The quantum mechanics gave us the sign that the elementary particles time is not related to our time by a real linear relationship as its in special relativity. Because special relativity can't accomplish transformation of time of elementary particles, otherwise we don't need to assume quantum mechanics postulates to describe their interactions. So we need the complex transformations to perform the definition of time of these particles. But if we stick with real one we can't define time and position of these particles by any logic language. This why you said its an assumption. Time can't be an assumption its reality and every particle is affected by time. May be position of these particles can be put under discussion. But time is difficult to be dismissed, because its related to the dynamic of the universe its even not just due to relative motion as in special relativity. This is only an approximation because there is no stationary particle in universe.  

Sadeem,

time is not an object and does not act on anything. It is the way to describe objects.

Regards,

Eugene.

This can give us a rough estimation of certain phenomenon which they call it probability in quantum mechanics. The problem if we want to understand the real physics we need to think deeper. The quantum mechanics as a whole can be reformulated in a classical manner and in a logical description using these complex transformations. The weirdness that may be appeared like the quantum Zeno effect and the non locality of entangled particles, single photon interference, the wave particle duality ....etc 

can all be interpreted using the classical language which involve locality and logic description. The measurement give you an impression at certain moment, but it can't describe the phenomenon from the particle side of view. This is the most important to think about; how the particles see each other and interact with each other not how we can see them. How we can see them can give you an impression about the interaction but can't give the real physics.

Dear Eugene

I'm not saying its an object, What I said its a reflection of particle's dynamic. It's not time that affects the particle, but its a measure to particle dynamics. You can consider it as a gauge of particle's velocity. But we can't understand time of elementary particles unless we understand their dynamics. Why there is non zero ground state energy of quantum particles, why photons are moving at speed of light? There is something that acts on them and give them these properties, what ever it be? the expansion of universe, the vacuum energy....etc 

But this triggers a complex dynamics of these particles that make them uncertain in their positions and time.

Among the various methods is also: "the wave function has no reality."  But we are speaking about quantum theory, that, as a theory, is mathematical.  And this theory is not consistent, therefore not predictive unless empirical elements of reality are put into it.

There is not even a precise definition of what a measurement is, so any resoning that uses this notion is void.

But we have nothing better now, Claude.

Sadeem,

time is the measure to describe dynamic. But if we are thinking about 4-dimensional objects as about reality, then there are no changes and dynamic at all. This is exactly what says relativity.

Regards,

Eugene.

Sadeem,

I do not understand what you are saying about uncertainty of position and time. Position times time does not give a measure of action so the quantisation of action does not give an uncertainty relationship. We may position a detector and time the ping with as much precision as our technology allows but we would then have no idea of the momentum and energy of the particle when it did arrive.

Time is not part of QM. The clock is external to the system and runs at exactly 60 seconds to the minute without any regard to what the quantum particle is doing.

We had nothing better than classical mechanics, we had nothing better than Aristotle theory.  Fortunately, not everybody said "shut-up and calculate, 'cause we have nothing better."  Many physicists are in the illusion that we got the absolute truth, from which we can say everything about Nature, like "the vacuum is unstable" and other drivels.  Physics is not about truth, it is about consistently modelling Nature.  I'm not a god, I'm a poor human.

Eugene,

if you depends on SR in analyzing dynamics you won't get anything, because it depends on relative motion. While it couldn't answer  the questions why the speed of light  is the limit speed  and cause of non zero gr state energy...etc so it can't be used to describe the dynamics of these particles.

Andy,

time is not just what your clock reads this only measured time I'm talking about the dynamic'related time which needs to understand the dynamics of elementary particles 

not just the measurement  process.