Why need *complex* wavefunction?

[jensa]

Thank you AEM for your reply, I appreciate it. And I will definitely check out the book you referred to sometime in the future when I have more time.

 

[DeepThought42]

The question is why is one single real number for example insufficient.

The real numbers are only a small portion of all numbers that exist. There is one single complex number which is a single point on the complex plane.

The real issue is a fault in the multiplication system.

 

[meopemuk]

I will look up that theory at some point.

OK, which dimension to use for the numbers can be justified there. But does it also justify why operators on vectors and scalar products for probabilities should be used? Sorry, if this question doesn't make sense. I'm good at QM at a "physicists level" only.

Yes, quantum logic provides a foundation for the entire formalism of quantum mechanics, including operators, vectors, scalar products etc. If you already know QM, you may find it very entertaining to see how all this machinery follows from a set of axioms, which are not much different from axioms of classical logic and probability. Actually, only one axiom of the standard Boolean logic needs to be modified - the distributive law.

Unfortunately, I don't know any good and comprehensive introductory level book or review on quantum logic. I got my knowledge from various places:

The original idea is more than 70 years old. This paper is well-written and highly recommended

G. Birkhoff, J. von Neumann, "The logic of quantum mechanics", Ann. Math., 37, (1936), 823

Possibly, the best elementary-level introduction can be found in Section 2-2 of

Mackey, G. W., The Mathematical foundations of Quantum Mechanics, W. A. Benjamin, Inc.
(1963), New York

Piron's book contains (the substantial part of) the proof of the theorem that I've mentioned. However, this book is a bit too heavy on the mathematical side.

Piron, C. Foundations of Quantum Physics, 1976, W. A. Benjamin Inc.
Reading, Massachusetts

 

[turin]

Can someone present a first order (in time) wave-equation which is not of the schrödinger form and does not have complex solutions?

The Majorana equation. The catch is that the wave function is a 4-component object.

 

[akhmeteli]

Yes, but what is a real :) point?
I do believe the underlying (but hidden) subject of this topic is a mysterious hatred of physicists versus the complex numbers, like, 'use see, we can do it without these bad complex numbers, which do not have any physical meaning!'

I don't want to speculate what other physicists love or hate. For me, the point is that this may be important for interpretation of quantum mechanics. For example, in the case of the de Broglie-Bohm interpretation, the electromagnetic field can play the role of the guiding field in the Klein-Gordon-Maxwell system (after you get rid of the complex wavefunction).

 

[jensa]

The Majorana equation. The catch is that the wave function is a 4-component object.

Hi turin, I appreciate your answer. I probably should have pointed out that it should also have to be a scalar wave-function. Anyway, it seems AEM has already called my bluff...

However, since you replied to memaybe I can ask you a question regarding the majorana equation? I am familiar with what we in the condensed matter community call majorana fermions, and I suppose that there should be a clear relationship.

In condensed matter community we can construct two (what we call) majorana (or real) fermions from a single complex one:

[tex]\chi_1=\psi+\psi^\dag, \quad \chi_2=i(\psi-\psi^\dag)[/tex]

with the properties [itex]\chi_i^\dag=\chi_i, \ i=1,2[/itex] (which together with anti-commutation relations of fermions yields interesting physics if one can separate spatially two majorana fermions from each other - has application to topological quantum computing)

The equation of motion (usually a complex one) for [itex]\psi[/itex] becomes a set of real coupled equations for [itex]\chi_i[/itex]. The analogy to what has been presented in this thread in terms of the Schrödinger equation should be obvious. This only replaces one complex equation to two real ones. In this sense majorana does not add anything useful to the discussion.


Going out on a limb on trying to argue the same for how high-energy physicists treat majorana fermions, I would start from the Wikipedia definition of Majorana equation:

[tex]i\partial \psi-m\psi_c=0[/tex]
(note: [itex]\partial[/itex] should be replaced with \partialslash which does not seem to work)
where [itex]\psi_c=\gamma^2\psi^*[/itex] i.e. it is related to [itex]\psi[/itex] through operation which includes complex conjugation. This equation does not seem to necessarily imply that its solutions must be real? Rather the special case where the field is (pseudo-)real [itex]\psi=\psi_c[/itex] is the case when its solutions are majorana fermions. (please correct me if this interpretation is wrong)

One could consider the above majorana equation together with its complex conjugate and form an equation for a spinor

[tex]\begin{pmatrix}\psi\\\psi_c\end{pmatrix}[/tex]

this spinor is also (pseudo-) real meaning that we can transform it into a basis where it is real. This whole procedure is analogous to producing from one complex spinor two real ones, or equivalently from one complex eqm to two real ones.

So it seems to me that either you postulate that [itex]\psi=\psi_c[/itex] or you use the procedure above to combine [itex]\psi[/itex] and [itex]\psi_c[/itex] into two real spinors, i.e. replacing one complex equation for [itex]\psi[/itex] into two real equations.

Am I making sense to you? My point is simply that I don't think majorana equation gives you any new insight into the problem than what has been posted in terms of replacing complex schrödinger equation with two real ones.

 

[jensa]

I am afraid I have repeated this several times in other threads, but this question does arise frequently. Not many people know about Shroedinger's very short article (Nature (1952), v.169, p.538), where he shows that quantum mechanics can actually do without complex numbers or, equivalently, two real numbers for the wave function, as, e.g., for any solution of the equations of the Klein-Gordon-Maxwell electrodynamics (a scalar charged field \psi interacting with electromagnetic field) there exists a physically equivalent solution with one real (not complex) field, which can be obtained from the original solution by a gauge transform (see some fine print in thread https://www.physicsforums.com/showthread.php?t=98603). Thus, the entire range of physical phenomena described by the Klein-Gordon-Maxwell electrodynamics may be described using real fields only. Shroedinger's comment: "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."

While I have not read this article and unfortunately don't have time (sorry for being lazy) I would still like to comment on this. This procedure seems very much like what is used in Higgs mechanism [note that I am in no way an expert on this]. An important point is that when you choose a gauge in this way it by no means removes the degree of freedom associated with the phase. Rather it is absorbed into the the gauge field. If you originally have a complex scalar field with two degrees of freedom (magnitude and phase) and a gauge field [itex]A^\mu[/itex] which, due to gauge invariance, has (only) three degrees of freedom, you will, after fixing the gauge in a way such that the scalar field becomes real, have one degree of freedom associated with the scalar field and 4! degrees of freedom of the gauge field.

So do I understand your point was that we could have taken the real field + non-gauge invariant EM field as a starting point? thus eliminating the need for complex fields at the "cost" of losing gauge invariance? This does seem quite appealing since gauge invariance is another one of those things which one might ask oneself "why do we need it?"

 

[RedX]

1) The Dirac equation can be made to be real as the gamma matrices can be made to be pure imaginary: http://en.wikipedia.org/wiki/Gamma_matrices#Majorana_basis

So if the differential equation has real coefficients, does that mean the solution does too?

2) The solution of the Majorana equation does not necessarily describe a Majorona particle. Only if solution is its own conjugate is the solution a Majorana particle. The Majorana equation is a perfectly good Lorentz-invariant equation: however, the Dirac equation is the one that matches experiment.

3) Imposing the condition that the field is self-conjugate is different than adding a field and its conjugate or subtracting a field and its conjugate (and multiplying by i). In the former case, you're solving for 1 Majorana particle, and in the latter case you're creating 2 Majorana particles from 1 Dirac particle.

 

[RedX]

Consider the path integral approach:

[tex] \int e^{iS}=<x',t'|x,t>[/tex]

Suppose at time t, the wavefunction [tex]\psi(x,t)[/tex] is real. Then at a later time t':

[tex] \psi(x',t')=\int e^{iS} \psi(x,t) [/tex]

So if the Lagrangian is real, then the action ought to be real, and the imaginary part comes form the i in the exponential. The i in the exponential is needed to keep the path integral from being infinity?

 

[akhmeteli]

So do I understand your point was that we could have taken the real field + non-gauge invariant EM field as a starting point? thus eliminating the need for complex fields at the "cost" of losing gauge invariance?

Yes.

 

[turin]

My point is simply that I don't think majorana equation gives you any new insight into the problem than what has been posted in terms of replacing complex schrödinger equation with two real ones.

Yes, I'm sorry. At the time I posted, I hadn't read all of the posts, so I was merely providing a first order equation and not worrying about how many components were needed. I don't have that book that AEM mentioned, but, in defense of the need for multiple components, I would say that, if you require the solution to the equation to be a scalar, and you require space to have more than 1 dimension, then you must introduce also some vector into the equation. So, while this can happen when the medium in which the wave propagates has a flow, I don't consider this a wave equation in the same sense of a matter-wave equation, in which the matter is supposed to be described as a wave that propagates through free space. So, basically, I must be misunderstanding the point.

 

[cesiumfrog]

Can someone present a first order (in time) wave-equation which is not of the schrödinger form and does not have complex solutions?

What's wrong with just the Schroedinger equation, replacing i with an operator that (rather than performing complex multiplication, and also different from the differential operators) factors any function into a Fourier basis then advances each component by a quarter wave? Doesn't this have solutions [itex]\phi(x,t)[/itex] that are real and single scalar valued?

 

[strangerep]

I just wanted to remind people that the power of complex numbers
(vs 2 real numbers) lies in the theory of complex-analytic (holomorphic)
functions, I.e., Cauchy-Riemann equations, and all that.

Holomorphic functions of a complex variable are not equivalent
(in general) to functions of 2 real variables.

This leads to immensely powerful theorems, used in advanced
physics quite a lot. (E.g., contour integration, Wick rotation,
path integrals, etc, etc.) The success of these tools suggests that
complex-analyticity does indeed have something important to do
with real world physics.

 

[RedX]

Holomorphic functions are indeed powerful, but have to do with certain complex functions of complex variables, while the wavefunction is a complex function of a real variable.

Wick's rotation seems to be just a mathematical trick to turn Minkowski space into Euclidean space so that you can integrate over hyperspheres, but in principle the rotation is not needed if you know how to do the integral. The path integral for scalar or vector fields gets the [tex] i\epsilon[/tex] from extracting the vacuum state, so I think it's just another trick (you could in principle integrate over the vacuum wavefunction to get your path integral instead of the epsilon prescription).

 

[samalkhaiat]

Does anyone know a deeper reason why the quantum mechanical wavefunction has to be complex?

In the non-relativistic QM, the complex nature of the wave function is as fundamental as the Plank constant itself.

1) If the dynamical equation EXPLICITLY contains the imaginary number i, as does Schroginger's equation, then a PAIR of REAL functions ( or their equivalent in the form of a SINGLE COMPLEX function) are needed for complete physical description.
If you insert [itex]\psi = f + ig[/itex] into Schrodinger's equation ([itex]2m=\hbar =1[/itex]):

[tex]i \partial_{t}\psi = - \nabla^{2}\psi[/tex]

you end up with 2 COUPLED equations

[tex]
\partial_{t}f + \nabla^{2}g = 0, \ \ \ \partial_{t}g - \nabla^{2}f =0
[/tex]

Thus we see that f and g are not INDEPENDENT of each other,i.e., neither of them alone is a solution of Schrodinger's equation.
Therefore, a complete solution requires the REAL pair (f,g) or their equivalent COMPLEX [itex]\psi[/itex].
Notice that the probability current

[tex]j_{i} = 2(f \partial_{i}g - g\partial_{i}f)[/tex]

is a mutual property of f and g, which vanishes when either is identically zero.

2) If the imaginary (i) does not show up explicitly in your equation, then the USE of COMPLEX function is just an auxiliary device.
Take the wave equation

[tex]\frac{\partial^{2}\phi}{\partial t^{2}} = \nabla^{2}\phi[/tex]

Now put [itex]\phi = f +ig[/itex]. This leads to the following two independent equations

[tex](\frac{\partial^{2}}{\partial t^{2}} - \nabla^{2})f = 0[/tex]

[tex](\frac{\partial^{2}}{\partial t^{2}} - \nabla^{2})g = 0[/tex]

Thus, f and g remain independent of each other,i.e., a complete solution can be given in terms of f alone or g alone. Therefore, in this case, the wave function [itex]\phi[/itex] need not necessarily be complex function.

3)You might now say this: But the first-order Schrodinger's equation can be replaced by the "equivalent" second-order equation

[tex]\frac{\partial^{2}\psi}{\partial t^{2}} + \nabla^{4}\psi = 0 \ \ \ (X)[/tex]

Therefore, according to (2), the wave function [itex]\psi[/itex] does not need to be a complex function! Does this mean that the complex nature of Schrodinger's wave function is not FUNDAMENTAL after all?
This quetions is, of course, meaningless, because it is based on a wrong conclusion. Let me explain this.
It is true (as explained in (2)) that [itex]\psi[/itex] in the second-order equation (X) does not need to be complex function. It is also true that the equation (X) is "equivalent" to the first-order Schrodinger's equation. IT IS, however, a WRONG equation to use in QM! One way to see this is to prove that NO CONSERVED PROBABILITY can be set up which depends on [itex]\psi[/itex] only and not on [itex]\partial_{t}\psi[/itex].
To do this, write

[tex]P = P(\psi)[/tex]

Then

[tex]\partial_{t} \int dx \ P(\psi) = \int dx \frac{\partial P}{\partial \psi} \frac{\partial \psi}{\partial t}[/tex]

If this expression is to vanish for arbitrary [itex]\psi[/itex], it is necessary that [itex]\partial_{t}\psi[/itex] be given in terms of [itex]\psi[/itex]. But this implies a FIRST-ORDER wave equation. In a SECOND-ORDER differential (like eq(X)), [itex]\partial_{t}\psi[/itex] can be given an arbitrary initial value. Therefore, the above expression cannot vanish for all [itex]\psi[/itex]. For example, if we put

[tex]\frac{\partial \psi}{\partial t} = \frac{\partial P}{\partial \psi}[/tex]

then

[tex]\frac{\partial}{\partial t} \int dx \ P(\psi) = \int dx \ (\partial P / \partial \psi )^{2} > 0[/tex]

So, we see that EQ(X) can not be PHYSICALLY equivalent to Schrodinger's equation.


regards

sam

 

[jensa]

ahhh...now THIS is the way to present the argument that I was trying to get at in my first post.

 

[turin]

2) If the imaginary (i) does not show up explicitly in your equation, then the USE of COMPLEX function is just an auxiliary device.

This is not obvious to me. We do know that this is not true for polynomials, so I wonder if it might not be true for diff eqs.

3)You might now say this: But the first-order Schrodinger's equation can be replaced by the "equivalent" second-order equation

[tex]\frac{\partial^{2}\psi}{\partial t^{2}} + \nabla^{4}\psi = 0 \ \ \ (X)[/tex]

It is also true that the equation (X) is "equivalent" to the first-order Schrodinger's equation.

This is not true. There are solutions to this equation that do not solve the free-particle Schroedinger equation (namely, the negative frequency solutions).

 

[PhilDSP]

The use of complex numbers is a relatively simple way to take advantage of symmetry relationships between 2 dimensions when expressed as a function of a third dimension.

In other words a wave function of 2 spatial dimensions measured in time can be expressed with complex math because the relationships are symmetric - the medium restores in exactly the same manner as it is stressed if you imagine a wave in an elastic medium.

Quaternions and quaternions reduced to spinors up the ante by taking an equivalent advantage of symmetry relationships between 3 spatial dimensions measured in time.

 

[turin]

I'm sorry for resurecting this thread, but something just occurred to me.

I think that the objection to the wavefunction having more than one component is like objecting to the vector potential in E&M. Of course, there are cases in E&M (i.e. just E-statics) in which the scalar potential is sufficient. And similarly, there are cases in QM (i.e. bound states) in which the wavefunction can be entirely real-valued. However, a real-scalar wavefunction simply doesn't work in general for QM, just as E&M cannot be generally described entirely in terms of the scalar potential.

A "real" version of Schroedinger's equation can be constructed using a 2-component wavefunction. Or, closer to the E&M analogy, the "real" Majorana equation can be solved by a 2-component spinor (if multiplication of the spinor components is anticommutative). I don't think this is any more unacceptable than the need for a vector potential in E&M. It just means that the field being described is not a scalar; it has some kind of direction associated with it. What's more, this direction represents Lorentz transformations in both the Maxwell and Majorana cases.

 

[Gerenuk]

Don't worry about resurrecting. Someone resurrected my very old thread here and I got some really good answers this time :)

Well, in E&M I can see why it is 3 components. It's because there are 3 directions in space.

Of course purely real QM is not compatible with the QM as we have it. The (philosophical) question is why we need something like the phase? So this question is heading into the direction of interpretations of quantum mechanics.

 

[turin]

... in E&M I can see why it is 3 components. It's because there are 3 directions in space.

Actually, E&M needs 4 components for the potential, and this is "because" space-time has 4 directions. I would say that E&M needs 4 components because the field is in the fundamental representation of the Lorentz group. This is actually the same reason why QM needs two components (ala Majorana), because the spinor is in a (unfaithful) representation of the Lorentz group.

Of course purely real QM is not compatible with the QM as we have it.

I don't understand what you mean by this. The Majorana equation is the "purely real QM" wave equation for a fermion. Maxwell's equations are the "purely real QM" wave equation for a photon. Unfortunately, we can't combine these two, because the Majorana fermion must be neutral, so perhaps that was your objection?

 

[Gerenuk]

Actually, E&M needs 4 components for the potential, and this is "because" space-time has 4 directions. I would say that E&M needs 4 components because the field is in the fundamental representation of the Lorentz group. This is actually the same reason why QM needs two components (ala Majorana), because the spinor is in a (unfaithful) representation of the Lorentz group.

I don't know enough to understand this (what's the best book *for physicist* to learn that? I only know very basic group theory), but I sense that is a sound reason.

I don't understand what you mean by this. The Majorana equation is the "purely real QM" wave equation for a fermion. Maxwell's equations are the "purely real QM" wave equation for a photon. Unfortunately, we can't combine these two, because the Majorana fermion must be neutral, so perhaps that was your objection?

In the initial question replace *complex* with *the structure with 2 dimensions and all the rules that belong to complex numbers*.

 

[turin]

what's the best book *for physicist* to learn that?

Sorry, I just seemed to sort of pick it up. Probably a good book on the standard model would discuss this, but I just took a course, and we didn't have a (single, specific) text, so I don't know what book I should recommend. If you look for standard model books, and then find one that you are considering to purchase, I can at least tell you if I hate it.

One of the basic principles in the standard model is that (fundamental) particles are represented by quanta of fields that are irreps of the Lorentz group (among other groups). The irrep is exactly the spin (and chirality) of the particle. So, for example, the Majorana equation describes spin-1/2 particles and Maxwell's equation(s) describes spin-1 particles.

In the initial question replace *complex* with *the structure with 2 dimensions and all the rules that belong to complex numbers*.

Ah, but this is indeed the case. I think someone has even posted in this thread that the complex numbers can be represented by a weighted sum of the 2x2 identity matrix and any single Pauli matrix. I don't know where to start in order to make the connection, so I will just suggest some features of the spin-1/2 particle (Majorana equation): it has two components, the Lie algebra of the Lorentz group acting on the spin-1/2 particle is (unfaithfully) represented with the Pauli matrices, the Dirac matrices are related to the Pauli matrices, the Majorana equation uses the real representation of the Dirac matrices.