Need clarification for mass dimensionality of fermion completeness relation

[Jay R. Yablon]

I am confused and must be missing something.

In three dimensional space, a Dirac spinor u has a mass dimension of 3/2.
This is clear in a number of ways:

The current four-vector J^u = psi-bar gamma^u psi has mass dimension = 3, with
each wavefunction psi-bar and psi carrying 3/2.

Further, because J^u = d^v d_v A^u (simplest form of Maxwell's equation in
covariant gauge), and A^u has mass dimension of 1, the J^u picks of two more
mass dimensions because of the second space derivative.

Further, the probability *density* = psi-bar^T* psi has to be 1/length^3 =
mass^3. (T* = conjugate transpose)

All of this I know is correct.

YET, when we take the Dirac completeness relation

SUM_spins u u-bar = p-dagger + m

it seems as if we are multiplying a together a column vector and a row vector
(u u-bar ) each with mass dimension 3/2, so should expect mass dimension of 3,
***but end up with p-dagger + m which has mass dimension of 1***.

Help! What am I missing?

Thanks,

Jay
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[Jos Bergervoet]

On 6/26/2012 4:56 PM, Jay R. Yablon wrote:
> I am confused and must be missing something.
>
> In three dimensional space, a Dirac spinor u has a mass dimension of 3/2.
> This is clear in a number of ways:
>
> The current four-vector J^u = psi-bar gamma^u psi has mass dimension = 3, with
> each wavefunction psi-bar and psi carrying 3/2.
>
> Further, because J^u = d^v d_v A^u (simplest form of Maxwell's equation in
> covariant gauge), and A^u has mass dimension of 1, the J^u picks of two more
> mass dimensions because of the second space derivative.
>
> Further, the probability *density* = psi-bar^T* psi has to be 1/length^3 =
> mass^3. (T* = conjugate transpose)
>
> All of this I know is correct.
>
> YET, when we take the Dirac completeness relation
>
> SUM_spins u u-bar = p-dagger + m
>
> it seems as if we are multiplying a together a column vector and a row vector
> (u u-bar ) each with mass dimension 3/2, so should expect mass dimension of 3,
> ***but end up with p-dagger + m which has mass dimension of 1***.
>
> Help! What am I missing?

In the completeness relation you work with
a set of "basis vectors" for Dirac spinors.
They apparently have dimension 1/2.

If you create a spinor wave function in
coordinate space you'll have to multiply the
basis vectors with coefficient functions of
x. These should then have dimension 1, to get
the dimension 3/2 that you want for the
spinor wave function. (Note that for a spinor
wave function in momentum space it is -3/2.)

--
Jos

[Jay R. Yablon]

In other words, we have to address this through the way in which we normalize the wayefunctions? Jay

[Jos Bergervoet]

On 7/3/2012 7:53 PM, Jay R. Yablon wrote:
> In other words, we have to address this through the way in which we normalize the wayefunctions? Jay

Well, the wavefunction requires dimensionality +3/2 in order to
get a dimensionless number its normalization integral (integral
d^3 x Psi_bar Psi). I don't think you'd want it otherwise..

But the four-spinor that is usually introduced for fermions is:
http://en.wikipedia.org/wiki/Dirac_spinor#Four-spinor_for_particles
which clearly has another dimension: +1/2. This could of course
have been done differently.

The problem is not typical for Dirac spinors. The non-relativistic
Schrodinger equation also requires a wave function of dimension
+3/2 for normalization, regardless whether it is a scalar or a
two-spinor. Still, in analyzing the two-spinors in SU(2) context,
we usually work with dimensionless two-vectors. So also there a
discrepancy, although now the difference is 3/2 instead of 1.

--
Jos