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Complex numbers and physical meaning

  1. Oct 25, 2015 #1
    I have to say that I am a bit confused with the use of complex numbers. I know that:
    1. They have been created by mathematicians to solve the "real"ly unsolved equation of x^2=-1.
    2. They are used in many aspects of physics, like waves and quantum theory, with terrific correspondence to the maths and experimental results.
    3. But when it comes to physical meaning, there is a gap. Tricks like squaring to find the probability width in order to get rid of the unwanted "imaginary" part, or just using the "real" part.

    I tried to find a physical explanation but with no lack. Specifically I would appreciate a physical correspondence in two aspects.

    A. The waveparticle solution of Schroedinger Equation.
    B. The Unit operator of time (evolution).

    Thanks for your time.
     
  2. jcsd
  3. Oct 25, 2015 #2
    I would say this is more a philosophical question, rather than scientific one, but interesting anyway! From my point of view, the complex numbers arise in too many parts of physics just as the only formalism needed to develop a theory. To explain it better, if you would try to develop some physical theories without them, they will end up being non-sense. The complex plane is an extension of the real numbers set, where more combinations of numbers are possible, and therefore, the possibilities of making a physical theory grow.

    In QM for example, all operators algebra is defined in complex algebraic form: Observable quantities are always real, and we know that hermitian operators have real eigenvalues always. If we would try to develop the theory just with real symmetric matrices, our "movements" would be very limited. The wave equation (Schrodinger) is a differential equation for a complex function, which by itself doesn't have any physical meaning because its complex. It is the magnitude square of this function. Why the wavefunction is complex arises from the definition of the hamiltonian operator: [tex]\hat{H}=i\hbar\partial_{t}[/tex]
    which is hermitian. This is a fascinating question and I hope some more people will join!
     
  4. Oct 25, 2015 #3

    bhobba

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    There is a deep reason complex numbers are necessary in QM:
    http://www.scottaaronson.com/democritus/lec9.html

    Just to show all roads lead to Rome its also fundamental to Schroedinger's equation:
    http://arxiv.org/pdf/1204.0653.pdf

    Almost forgot to mention complex numbers are required for a very important theorem - called Wigners Theorem - see attached file

    Thanks
    Bill
     

    Attached Files:

    Last edited: Oct 25, 2015
  5. Oct 26, 2015 #4
    Thanks for your time.

    Excuse me if I didn't understand correctly but again I received reasons and necessity of using complex numbers and not physical explanation!

    I think, I have already stated that they seem a necessity and they help with the maths. I am not questioning that, nor I am searching for a different way, I am just confused with their physical correspondence. In every lecture I have watched or read, they just overpass the physical justification and they just prove the maths. As far as I am concerned there is an obligation as physics teacher to give a physical meaning of everything I am using. If not then there is a gap, and it is not a philosophical query. It is a deep gap in our understanding of the maths and equations that we are using.

    We cannot use a mathematical tool without explaining it's physical correspondence. The addition is putting together, multiplication is growing by a number, division is splitting, derivative of space in time is the velocity, the second derivative is the acceleration, the curvature of the gravitational field is the mass, ω and k are the circular frequency and the wave number respectively, and so on. What is the meaning of an imaginary number in physics? Not its squared product, nor its "real" part.

    I am trying to find out if there is any explanation out there that I am missing, and until now, nobody has given me a satisfying answer. I hoped somebody would enlighten me.
     
    Last edited: Oct 26, 2015
  6. Oct 26, 2015 #5

    PeroK

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    I don't know that much QM, but I can give you a mathematical perspective.

    First, the term "imaginary" has no significance. Also, that complex numbers are called "numbers" has no significance. They could have been called, perhaps, "multiplicative vectors". They are a mathematical tool. A lot of people seem to get worked up about complex numbers and their physical "existence", while not worrying about the physical existence of an ##n \times n## matrix, for example.

    A wave function is a complex function of a two real variables ##x, t##. That means that the wave function is two dimensional. So, you can write:

    ##\Psi (x, t) = \Psi_1(x, t) + i \Psi_2 (x, t)##

    If you put this into the Schrodinger equation, you can generate a pair of simultaneous differential equations:

    ##\hbar \frac{\partial \Psi_1}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi_2}{\partial x^2} + V \Psi_2##

    ##-\hbar \frac{\partial \Psi_2}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi_1}{\partial x^2} + V \Psi_1##

    This pair of equations is mathematically equivalent to the Schrodinger equation and is devoid of complex numbers. Perhaps your question, therefore, boils down to the physical significance of the wave function comprising two interdependent (real) wave functions.
     
  7. Oct 26, 2015 #6
    The reason is invariance under time reversal. The consequence is that there are always two degenerate solutions that are 90 degrees out of phase. This twofold degeneracy is conveniently described with complex numbers.
     
  8. Oct 26, 2015 #7

    bhobba

    Staff: Mentor

    Since physics is written in the language of math I am not sure how you separate a math explanation from a physical one. Wanting an explanation on your terms is not a good idea because nature may not oblige. Its better to accept nature as is. If that means the explanation is what you call mathematical rather than physical then thats the way it is.

    Anyway here is another take on the issue:
    http://www.scottaaronson.com/papers/island.pdf

    See point 2.

    Thanks
    Bill
     
  9. Oct 26, 2015 #8

    bhobba

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    You are of course correct.

    Such questions really are - why is QM most elegantly expressed using complex numbers.

    Looking at it that way it's quite reasonable the answer is mathematical rather than physical - not that its easy to separate the two.

    Thanks
    Bill
     
    Last edited: Oct 29, 2015
  10. Oct 26, 2015 #9

    PeroK

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    The most elegant thing that complex numbers do, IMO, is the way they unify the exponential function with the simple harmonic functions (sin and cos). Looking at the second question (B) in the OP, the "imaginary" exponential function is an elegant way to express two simple harmonics 90° out of phase.
     
    Last edited: Oct 26, 2015
  11. Oct 26, 2015 #10
    Speaking strictly as a layman, this is how I visualize it: The probability amplitudes that are calculated in QM consist of an absolute value multiplied by a phase angle. When probability amplitudes are added, as in a superposition, they must be added vectorially, taking phase angle into account, before being squared to find an actual physical probability. Complex numbers enter the picture because they are a traditional, convenient way to express the phase angle in terms of its x and y components, the cosine and sine. Senior posters please correct me.
     
  12. Oct 27, 2015 #11

    zonde

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    I would like to give shorter version of what PeroK said that does not look like a math:
    Imaginary number corresponds to two real quantities that are interdependent in a certain way in respect to final effect.
     
  13. Oct 27, 2015 #12
    This is a slightly reversed take on it.. In analogy to the situation of analytical mechanics where one requires stipulation of the generalised coordinates and generalised velocities such that the total set of these quantities constitutes the independent dynamical variables, the phase θ(x, t) and the amplitude R(x, t) of the wavefunction must be specified at any given time for every point in space. In effect, the phase and the amplitude serve as the independent dynamical variables for quantum mechanical systems. This is clearly evident in view of the Schrodinger equation being a first order complex partial differential equation with respect to time. At first sight, one may perceive a significant structural difference between the mechanics of classical and quantum systems given that the former seemingly evolve in time according to a set of n second order Euler-Lagrange equations which require specification of quantities which are zeroth order and first order in time. However, the dynamics of classical mechanical systems occur in a phase space formed by the 2n coordinates (q, p) which are zeroth order in time ”positional” type coordinates, analogous
    to the situation for quantum mechanical systems. In an identical fashion to the classical domain where the canonical momenta encode the flow of particles and are determined by the normal vectors of surfaces of constant action, quantum mechanical flow of probability is determined by local phase differences and directed normal to surfaces of constant phase. Accordingly, the requirement of specifying generalised velocity variables in the Euler-Lagrange formalism which are first order in time, translates to specifying a distribution of the phase of the wavefunction over space in the quantum realm, a quantity which is zeroth order in time. This is in complete analogy with the passage from the Lagrangian to Hamiltonian formalism of mechanics. So given that the S.E is first order in time and Newtonian mechanics is governed by a second order equation, 2 zeroth order dynamical variables are required: one quantity representing the "initial coordinates" i.e. the amplitude function and another quantity representing the "initial" velocities i.e. the phase function.
     
  14. Oct 27, 2015 #13

    Urs Schreiber

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    Here is one way to see it:

    First of all, a careful inspection shows that action functionals in physics have to be regarded as taking values not in the real numbers, as sometimes assumed, but in the real numbers modulo the integers. (This is discussed in the "PF-Insights"-article titled Higher prequantum geometry II: The global action functional.)

    The first implication of this is that the covariant phase space of a field theory carries a principal bundle for the circle group U(1) = R/Z. (This is the content of Higher prequantum geometry IV: The covariant phase space. For more background see also the exposition at fiber bundles in physics).

    Now quantum wave functions are the sections of a linearization of this U(1)-principal bundle on phase space. Such a linearization is given by a choice of ring R and a group homomorphism U(1) -> GL_1(R) from the circle group to the group of invertible elements of R.

    Hence the question comes down to what choices of rings R there is such that U(1) canonically maps to their multiplicative groups of invertible elements.

    The choice R = C the complex numbers is the canonical such choice.
     
  15. Oct 27, 2015 #14
    After Urs's post the following may seem a bit trivial :p, but may contribute in some way. The canonical equations of motion show that the Hamiltonian function serves as a generating function for time translations of the independent dynamical variables. The simplectic structure of these ## 2n ## equations and their appearance in pairs, can be cast in the form of a single set ## n ## if the paired conjugate variables are replaced by complex variables: $$ c_k = \frac{1}{\sqrt{2}} (q_k +ip_k) $$ $$ c_k^* = \frac{1}{\sqrt{2}} (q_k -ip_k) $$. With this transformation, we now have ## n ## complex canonical equations: $$ i\frac{d}{dt}c_k =\frac{\partial{H}}{\partial{c_k^*}} $$. Come to think of it, is this actually why we call it "phase space"?
     
  16. Oct 27, 2015 #15

    Urs Schreiber

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    What this is pointing out is that the phase space ##\mathbb{R}^{2n}## has, apart from its standard "real polarization" also a complex polarization (see at polarization). However, not every phase space admits a complex polarization (locally it does, but globally there may be obstructions). So, while complex polarizations are intimately related to the complex phase of wave functions, the occasional existence of complex polarizations does not seem to really answer the question as to why, fundamentally, wave functions have complex phases, I would think.

    Yeah, this is something one should wonder about at least once in a life. I once wondered about this, too. But the history was different. The short answer is that Boltzmann thaught of a point in phase space as a specifying "phase of the motion" of a gas in a box, and that's where the term more or less comes from. But the long answer is much longer, much more convoluted, in fact somewhat weird, also entertaining, and in any case a good reminder for anyone who believes that the scientific community as a whole operates in a rational manner: see
    • David Nolte, The tangled tale of phase space, PhysicsToday (April 2010) (pdf)
     
  17. Oct 27, 2015 #16
    Thank you Urs for a very informative reply! When you say
    do you intend that there are situations which occur with gauge field singularities or something? Preventing analysis of the connection between different complex polarizations at different points of the cotangent bundle? Apologies for my lack of knowledge in differential forms..
     
  18. Oct 27, 2015 #17

    Urs Schreiber

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    First of all, phase spaces need in general not be cotangent bundles. Even cotangent bundles in general only carry almost complex structure, not necessarily complex structure.
     
  19. Oct 27, 2015 #18
    Actually, you can question that (please see https://www.physicsforums.com/threads/why-are-qm-wave-functions-complex.683821/#post-4337679 )
     
  20. Oct 28, 2015 #19

    Urs Schreiber

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    There does not seem to be any room to question that quantum states in general have complex phases. That you may find one with a real phase is not proof to the contrary.

    And this is precisely due to the obvious physical interpretation that the OP is after: a space-dependent complex phase of a wave function is the way that momentum is encoded by the wave function. A wave function of a free particle of fixed momentum in flat space is nothing but a complex phase, linearly varying in space, and this is at the very heart of the nature of quantum physics, the uncertainty relation, and all those things that were experimentally verified and understood 100 years back.
     
  21. Oct 28, 2015 #20

    Khashishi

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    There are two philosophical stances you can take here.
    1. Numbers are tools we use in various models to describe nature.
    2. Numbers are real things that we discover.
    I choose to accept 1. Complex numbers are just a different and more general tool than real numbers to use in our models. And there are plenty of models that use complex numbers. Statement 2 raises all sorts of questions: what kind of numbers are "real"? Just whole numbers? Real numbers? I don't think these questions can be answered. But statement 1 is easy to come to grips with.
     
  22. Oct 28, 2015 #21
    It seems that Schroedinger had a different opinion on this issue. In his article (Nature, v.169, p.538(1952)) he wrote, discussing the Klein-Gordon-Maxwell electrodynamics: "That the wave function ... can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about 'charged' fields requiring complex representation." And he did not have in mind a replacement of a complex function by two real ones. Of course, even Schroedinger could be wrong, but then, I guess, you should explain where and why his published article was wrong. Let me note that wave function's phase is not gauge-independent, so it does not have any definite sense until you've fixed the gauge. Let me add that you can make any scalar wave function real by a gauge transform (at least, locally). Moreover, you can use just one real function instead of the Dirac spinor function, as I showed in my article in the Journal of Mathematical Physics (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf)

    There is no contradiction between what I wrote above and "the uncertainty relation, and all those things that were experimentally verified and understood 100 years back." I use the standard Klein-Gordon and Dirac equations of quantum mechanics.
     
  23. Oct 28, 2015 #22
    Ok..I have much reading to do then! Are a set of n coordinates ("momenta" usually but given arbitrariness under canonical transformations, I'll just say n coordinates) taking on values in the cotangent spaces not a requirement to ensure an invariant differential form of the action?: $$dA=\sum_i p_i dq_i$$
     
  24. Oct 28, 2015 #23
    Was the point not considered in Urs's earlier post, that action functionals take on values in the real numbers modulo the integers (conjugate action-angle variables pairs). The point being that action twists around from point to point and flow occurs along directions of non-zero action/phase twist? This is aside from the fact that one needs to also specify a second scalar function i.e amplitude function, serving as an initial or boundary "particle distribution".
     
    Last edited: Oct 28, 2015
  25. Oct 28, 2015 #24
    I'm guessing a choice of gauge was presumed and doesn't seem (at first sight anyway) to affect the underlying issue, the gauge field just curves the space over which flow (canonical momenta) occurs and simply needs to be accounted for. As an analogy, we could still ask about the structure of line elements in Minkowski space without needing to specify the general Riemannian geometry of spacetime.
     
  26. Oct 28, 2015 #25
    I'm unsure of whether the OP was looking for a reason for the use of Complex numbers as resulting from a required mathematical structure underpinning a physical theory or whether he was looking for some direct correspondence of complex numbers to physical reality. I usually presume people mean the former when they ask a question of this type. Any theory, be it physical or mathematical, remains an abstraction of reality and the idea that it is in direct correspondence with reality or that it ##\textit{is}## the reality, seems to me anyway like a rather limited and incoherent view. So.. yeah, guessing the OP's question is more geared towards the former.

    If it was the latter then I would say numbers and theories are (in ref to your second statement) things we create, which have relevance within a certain domain of experience of reality, if any at all. A map is an abstraction of a city which helps guide you through a city, yet there is evidently no part of the map which corresponds to the city. If it's a bad map you notice incoherence which becomes clear when you are not getting the result you expected (maybe you've just walked into a river) and you need a new map things, if it's a good map it will guide you well, so well sometimes that you may even start to believe the map is in direct correspondence with the city. When you start to probe things a bit more and ask more and more subtle questions, you realise you're continuously having to remap how things appear to you and that it wasn't that the old good maps were "wrong", but simply that there were limits to their relevance of our experience of reality.
    The question of a direct correspondence with reality of a number or theory seems, in my view, nonsensical in the first place given that they are abstractions of reality.
     
    Last edited: Oct 28, 2015
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