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Science Forum Index » Physics - Research Forum » Several questions about Dirac's equation and Dirac...
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| Jay R. Yablon... |
 Posted: Sun May 11, 2008 10:08 am |
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To all,
I have several queries about the Dirac equation and associated Dirac
spinors psi and quantum field theory (QFT).
1. In units hbar=c=1, a Dirac spinor psi has mass dimension +3/2. Yet,
the "inside" of a Dirac spinor for, say, a spin up electron is often
written as (transposed, easier for ASCII):
(1 0 p_z/(E+m) P_+/(E+m) ) (1)
which is dimensionless, and the normalization factor is written as:
sqrt [(E+m)/2m] or sqrt(E+m) (2)
The latter of (2) at least has mass dimension of 1/2, but what is the
representation (or normalization) of this which explicitly shows the
+3/2 mass dimensionality? Do we just multiply (1) through by m? Or by
E+m? What is the normalization for psi*T psi with mass dimension +3
which drives this? *T=conjugate transpose.
2. The Dirac spinor in compact form is often written as:
psi = u(p) exp [-i p x] (3)
I understand what happens from there, but why start with a complex plane
wave? Why not, for example, start with a more general form:
psi = u(p) exp [-(1/2)Ax^2 + Bx + V(x)] (4)
where V(x) is a general polynomial in x? Aside from the math maybe
being more difficult, would there be anything wrong with obtaining
solutions to Dirac's equation:
(i gamma^u d_u - m) psi (5)
using the more general waveform (4)?
3. If (4) is a valid general waveform, and recognizing that to keep the
exponent dimensionless, A must have mass dimension 2 and B mass
dimension 1, is it reasonable to suppose that A=m^2 where m is the
electron mass, and that B=p is the momentum (or B=E the energy) for the
time dependence of the wavefunction?
4. The "central identity of quantum field theory" as it is called by
Zee (see his page 167), for a quantum field psi, is given by:
$Dpsi exp[-(1/2)A psi^2 + B^psi +V(psi)
= exp[(1/2)B^2/A -V(d/dJ)] (6)
(I have changed his K-->A and J-->B to contrast with (4).) Let's focus
on V. As Zee puts it: "we single out the term quadratic in psi . . .
and call the rest V(psi). This is just another way of saying that V is
a generalized polynomial in psi, and, depending on the (unspecified)
coefficients of each order can be ay function under the sun. Being in
the exponent, V must be dimensionless. As a possible point of
reference, let me also point out my one-page calculation, earlier
posted, at
http://jayryablon.files.wordpress.com/2008/05/zee-baby-problem.pdf.
(Download and open if left click does not work.) Several questions:
A) V is often used to represent a potential. Yet a potential has mass
dimension of 1. Is V to be understood in any context as a potential?
(It seems somewhat apparent that it is.) If so, how does one get from
mass dimension 0 to mass dimension +1?
B) What is the most straightforward physical interpretation of V in
(6)? What would be the most straightforward interpretation of V in (4)?
C) If V in either or both of (4) and (6) is to be interpreted as a
potential with mass dimension of +1, then where is the connection to
gauge symmetry? That is, usually a potential comes about as the "0"
component of the potential 4-vector A^u, which in turn enters through
imposing gauge symmetry. How would one show V in (4) and (6) as arising
from gauge symmetry, as the "0" component of A^u?
Thanks,
Jay.
____________________________
Jay R. Yablon
Email: jyablon at (no spam) nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm |
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