Re: Why no tensors in quantum mechanics?

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

.

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