Re: please help my confusion about particles and irreps.

On 2005-05-28, Eugene Stefanovich <eugenev@xxxxxxxxxxxx> wrote:
> Igor Khavkine wrote:
>> On 2005-05-26, Eugene Stefanovich <eugenev@xxxxxxxxxxxx> wrote:

>>>Another thing is Dirac's electron-positron quantum field which is
>>>a 4-component operator function on the 4-dimensional Minkowski
>>>space. This function has no relationship to the measured probability
>>>amplitude. It has no relationship to the description of the state of
>>>the system in QFT.
>>>It plays a role of a "building block" for interactions in
>>>QED (or Standard model). U(1) symmetry refers to the quantum field,
>>>but not to the electron's wave function.
>>
>> The two are not completely unrelated. Again, referring to the last
>> chapter of Weinberg's first volume. If PSI(x) is the Dirac field
>> operator, then psi+(x) = <0|PSI(x)|1 particle> and
>> psi-(x) = <1 particle|PSI(x)|0> are respectively positive and negative
>> frequency solutions of the Dirac equation. On the other hand, in the
>> non-relativistic limit, two of the components of the Dirac spinor psi(x)
>> give the Schroedinger wave function.
>>
>> So, many of the things that can be said about the Dirac field operator
>> PSI(x) can be said about the Dirac and Schroedinger wave functions. For
>> instance it carries a representation of the U(1) gauge group, PSI(x) ->
>> exp(i e theta) PSI(x), and in the same way does psi(x) and its
>> non-relativistic limit.
>
> I do not think that Dirac's 4-component spinors provide a satisfactory
> description of 1-electron states. Zitterbewegung, "Klein paradox", etc.
> (see Bjorken & Drell, vol. 1) do not look like acceptable physical
> properties.

The great thing about the equivalence of the Dirac wave function picture
and the 1-particle sector of the Fock space picture is that they are,
well, equivalent. So any effects that show up in the Dirac wave function
picture also show up, in a suitable guise, in the Fock space
representation of the quantum field theory.

Zitterbewegung is related to the superposition of particle and
antiparticle states in the 1-particle sector. The Klein paradox and
other effects in very strong electromagnetic fields are related to a
different choice of the field theory ground state, compared to the
situation where the strong field is absent.

Of course, the mere presence of these effects in the field theory
description does not automatically endow them with a physical
interpretation. I am dubious about the importance of Zitterbewegung as,
to my knowledge, particle-antiparticle superpositions of indefinite
charge have never been observed. I cannot say that I understand the
implications of the effects central to Klein's so-called paradox, but
that hasn't stopped people who understand these things better from
performing experiments and calculations where these supercritical E&M
fields are realized.

Reinhardt, M\"uller, Greiner. Phys. Rev. A, v.24 p.103 (1981)

I know little more about this subject, but I am sure there has been more
progress since the '80s.

> On the other hand,
> Wigner's description of 1-particle states by vectors in the space of
> an unitary irreducible representation of the Poincare group satisfies all
> desirable physical properties. Electron's wave function in this
> representation is a 2-component function on the momentum (or position)
> space psi(p, sigma) (or psi(r, sigma)) ,
> where sigma = -1/2, 1/2 are two possible spin projections.
> These functions have clear probabilistic interpretation, e.g.,
>
> \int_V \sum_{sigma} |psi(p, sigma)|^2 dp
>
> provides the probability of finding the particle in the region V of
> the momentum space.
>
> \int_W \sum_{sigma} |psi(r, sigma)|^2 dr
>
> is the probability of finding the particle in the region W of
> the position space. The operators of momentum p, energy
> h= sqrt(m^2 + p^2), and velocity v = p/h
> have simple physically transparent form in this representation
> (this is not true for Dirac's "wave functions"). The position
> operator is easily obtained as well. The time evolution of
> Wigner's wave functions is described by obvious relativistic
> generalization of the Schroedinger equation
>
> -i d/dt psi (t) = sqrt(m^2 + p^2) psi (t)
>
> So, my suggestion is to use Wigner's functions to describe 1-electron
> states and to keep Dirac's 4-component quantum fields as formal
> linear combinations of creation and annihilation operators that are
> used for building interaction operators in the Fock space. Then
> Dirac's equation is a formal relationship satisfied by these formal
> fields, and should not be interpreted as a relativistic analogue of
> the Schroedinger equation.
> If you follow this suggestion, then the physical interpretation
> of the theory is clear and unambiguous.

Unless I misunderstand, your psi(p,s) is the coefficient of the
state of definite momentum and spin in the expansion of a given
1-particle state:

|psi> = sum_s int_p dp psi(p,s) |p,s>.

Since the state |psi> can be embedded in the space of Dirac wave
functions using the matrix elements <0|PSI(x)|psi> and <psi|PSI(x)|0>,
with the inner product carried over by the field operator
anti-commutation relations, all the operations that you described above
can be performed on Dirac wave functions.

The fact that you find the description you give above preferable, does
not invalidate alternatives nor stop anyone from using them without many
complaints.

Igor

.

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