Why need *complex* wavefunction?

[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

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

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.