For test purposes I wanted to force an inter-member LOCK dependency. This turns out to be pretty easy. First I created a table with a couple of dummy rows
db2 create table x '(x int)' db2 insert into x 'values(1)' db2 insert into x 'values(2)' db2 insert into x 'values(3)' db2 commit
Then setup an uncommitted update to these rows
# member 1 db2 connect to testdb2 db2 +c update x set x = x + 1
Doing the same thing on another member hangs as expected:
# member 2 db2 connect to testdb2 db2 +c update x set x = x + 1
commit on the first member resolves the hang. Interestingly one can do a ‘select * from x’ without any trouble, since we allow concurrent read and update, but one gets the pre-commit values of the rows in the table.
I had seen, but not the 
matrices, nor the 
notation. Strangerep in 
matrices as suggested.
. We want 
. How about 
? For that matrix we have
![\begin{aligned}\sigma^{\mu\nu} = \frac{i}{2} \left[{\gamma^\mu},{\gamma^\nu}\right]\end{aligned} \hspace{\stretch{1}}(3.8)](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Csigma%5E%7B%5Cmu%5Cnu%7D+%3D+%5Cfrac%7Bi%7D%7B2%7D+%5Cleft%5B%7B%5Cgamma%5E%5Cmu%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D%5Cend%7Baligned%7D+%5Chspace%7B%5Cstretch%7B1%7D%7D%283.8%29&bg=fafcff&fg=2a2a2a&s=0 "\begin{aligned}\sigma^{\mu\nu} = \frac{i}{2} \left[{\gamma^\mu},{\gamma^\nu}\right]\end{aligned} \hspace{\stretch{1}}(3.8)")
and 
. That first is
is zero, and for 
is
is 
.
in (1.26), (1.31), and (1.33) to a dagger.
instead of 
used.
s omitted after first equality.
missing.
operators from Eq. (3.58)
should be 
.
on wrong side of 
in the exponent.
is missing before the integral. Note that without this (4.67) appears incorrect (off by a factor of 
, but the error is really just in (4.65).
.
is not a solution to (4.122). This should be 
and (4.126) should be 
. This fixes the apparent error in sign in equations 4.129 and 4.130 which are correct as is.
.
. 
is missing prime on the 
index.
is in bold.
, and 
s aren’t in bold like 
in the kets.
.
isn’t in bold.
is in bold.![[S_i]](http://s0.wp.com/latex.php?latex=%5BS_i%5D&bg=fafcff&fg=2a2a2a&s=0 "[S_i]")
used instead of 
.
in bold. 
not in bold.
factor missing on RHS.
, but call these mechanical momenta on prev page.
s are in bold.
is constant, leaving it unclear how the gauge condition and how the curl of 
reproduces 
.
should be 
.
. ie: Clarify bounds, and add a factor of 
in the denominator which is required for the cancellation of (6.82).
s.
.
should be negated.
not 
.
factor inside parens.
instead of 
in expression for 
.
.
should be 
in the braket gives![\begin{aligned}{\langle {x} \rvert} e^{\frac{i}{\hbar}(p_0 X - x_0 P)} {\lvert {0} \rangle}&={\langle {x} \rvert} e^{\frac{i}{\hbar}p_0 X }e^{-\frac{i}{\hbar}x_0 P}e^{-\frac{i}{2\hbar}x_0 p_0 \left[{X},{P}\right]}{\lvert {0} \rangle} \\ &={\langle {x} \rvert} e^{\frac{i}{\hbar}p_0 X }e^{-\frac{i}{\hbar}x_0 P}e^{\frac{x_0 p_0}{2} }{\lvert {0} \rangle} \\ &=e^{\frac{x_0 p_0}{2} }{\langle {x} \rvert} e^{\frac{i}{\hbar}p_0 X} e^{-\frac{i}{\hbar}x_0 P}{\lvert {0} \rangle} \\ &=e^{\frac{x_0 p_0}{2} }\left({\langle {0} \rvert} e^{\frac{i}{\hbar}x_0 P}e^{-\frac{i}{\hbar}p_0 X} {\lvert {x} \rangle}\right)^{*} \\ &=e^{\frac{x_0 p_0}{2} }\left({\langle {0} \rvert} e^{\frac{i}{\hbar}x_0 P}{\lvert {x} \rangle}e^{-\frac{i}{\hbar}p_0 x} \right)^{*} \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \left({\langle {0} \rvert} e^{\frac{i}{\hbar}x_0 P}{\lvert {x} \rangle}\right)^{*} \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \left(\left\langle{{0}} \vert {{x - x_0}}\right\rangle\right)^{*} \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \left\langle{{x - x_0}} \vert {{0}}\right\rangle \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \psi_0(x - x_0, 0)\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%7B%5Clangle+%7Bx%7D+%5Crvert%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7D%28p_0+X+-+x_0+P%29%7D+%7B%5Clvert+%7B0%7D+%5Crangle%7D%26%3D%7B%5Clangle+%7Bx%7D+%5Crvert%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7Dp_0+X+%7De%5E%7B-%5Cfrac%7Bi%7D%7B%5Chbar%7Dx_0+P%7De%5E%7B-%5Cfrac%7Bi%7D%7B2%5Chbar%7Dx_0+p_0+%5Cleft%5B%7BX%7D%2C%7BP%7D%5Cright%5D%7D%7B%5Clvert+%7B0%7D+%5Crangle%7D+%5C%5C+%26%3D%7B%5Clangle+%7Bx%7D+%5Crvert%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7Dp_0+X+%7De%5E%7B-%5Cfrac%7Bi%7D%7B%5Chbar%7Dx_0+P%7De%5E%7B%5Cfrac%7Bx_0+p_0%7D%7B2%7D+%7D%7B%5Clvert+%7B0%7D+%5Crangle%7D+%5C%5C+%26%3De%5E%7B%5Cfrac%7Bx_0+p_0%7D%7B2%7D+%7D%7B%5Clangle+%7Bx%7D+%5Crvert%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7Dp_0+X%7D+e%5E%7B-%5Cfrac%7Bi%7D%7B%5Chbar%7Dx_0+P%7D%7B%5Clvert+%7B0%7D+%5Crangle%7D+%5C%5C+%26%3De%5E%7B%5Cfrac%7Bx_0+p_0%7D%7B2%7D+%7D%5Cleft%28%7B%5Clangle+%7B0%7D+%5Crvert%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7Dx_0+P%7De%5E%7B-%5Cfrac%7Bi%7D%7B%5Chbar%7Dp_0+X%7D+%7B%5Clvert+%7Bx%7D+%5Crangle%7D%5Cright%29%5E%7B%2A%7D+%5C%5C+%26%3De%5E%7B%5Cfrac%7Bx_0+p_0%7D%7B2%7D+%7D%5Cleft%28%7B%5Clangle+%7B0%7D+%5Crvert%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7Dx_0+P%7D%7B%5Clvert+%7Bx%7D+%5Crangle%7De%5E%7B-%5Cfrac%7Bi%7D%7B%5Chbar%7Dp_0+x%7D+%5Cright%29%5E%7B%2A%7D+%5C%5C+%26%3De%5E%7B%5Cfrac%7Bx_0+p_0%7D%7B2%7D+%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7Dp_0+x%7D+%5Cleft%28%7B%5Clangle+%7B0%7D+%5Crvert%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7Dx_0+P%7D%7B%5Clvert+%7Bx%7D+%5Crangle%7D%5Cright%29%5E%7B%2A%7D+%5C%5C+%26%3De%5E%7B%5Cfrac%7Bx_0+p_0%7D%7B2%7D+%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7Dp_0+x%7D+%5Cleft%28%5Cleft%5Clangle%7B%7B0%7D%7D+%5Cvert+%7B%7Bx+-+x_0%7D%7D%5Cright%5Crangle%5Cright%29%5E%7B%2A%7D+%5C%5C+%26%3De%5E%7B%5Cfrac%7Bx_0+p_0%7D%7B2%7D+%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7Dp_0+x%7D+%5Cleft%5Clangle%7B%7Bx+-+x_0%7D%7D+%5Cvert+%7B%7B0%7D%7D%5Cright%5Crangle+%5C%5C+%26%3De%5E%7B%5Cfrac%7Bx_0+p_0%7D%7B2%7D+%7D+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7Dp_0+x%7D+%5Cpsi_0%28x+-+x_0%2C+0%29%5Cend%7Baligned%7D+&bg=fafcff&fg=2a2a2a&s=0 "\begin{aligned}{\langle {x} \rvert} e^{\frac{i}{\hbar}(p_0 X - x_0 P)} {\lvert {0} \rangle}&={\langle {x} \rvert} e^{\frac{i}{\hbar}p_0 X }e^{-\frac{i}{\hbar}x_0 P}e^{-\frac{i}{2\hbar}x_0 p_0 \left[{X},{P}\right]}{\lvert {0} \rangle} \\ &={\langle {x} \rvert} e^{\frac{i}{\hbar}p_0 X }e^{-\frac{i}{\hbar}x_0 P}e^{\frac{x_0 p_0}{2} }{\lvert {0} \rangle} \\ &=e^{\frac{x_0 p_0}{2} }{\langle {x} \rvert} e^{\frac{i}{\hbar}p_0 X} e^{-\frac{i}{\hbar}x_0 P}{\lvert {0} \rangle} \\ &=e^{\frac{x_0 p_0}{2} }\left({\langle {0} \rvert} e^{\frac{i}{\hbar}x_0 P}e^{-\frac{i}{\hbar}p_0 X} {\lvert {x} \rangle}\right)^{*} \\ &=e^{\frac{x_0 p_0}{2} }\left({\langle {0} \rvert} e^{\frac{i}{\hbar}x_0 P}{\lvert {x} \rangle}e^{-\frac{i}{\hbar}p_0 x} \right)^{*} \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \left({\langle {0} \rvert} e^{\frac{i}{\hbar}x_0 P}{\lvert {x} \rangle}\right)^{*} \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \left(\left\langle{{0}} \vert {{x - x_0}}\right\rangle\right)^{*} \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \left\langle{{x - x_0}} \vert {{0}}\right\rangle \\ &=e^{\frac{x_0 p_0}{2} } e^{\frac{i}{\hbar}p_0 x} \psi_0(x - x_0, 0)\end{aligned}")
multiplying the wave function. Because of this I think that (10.51) should be a proportionality statement, and not an equality as in
ought to have braces and read 
.
term from the integral.
is given for a mixed tensor representation. This is 
. The other mixed representation 
transforms as 
.
written instead of 
instead of 
not 
.
is the energy of the particle, not 
. There’s also an 
.
presumed constant ought to be incorporated into 
. (I’ve added approximately equal for the second part since that wasn’t specified which I found confusing).
on second like of the change of variables.
instead of 
.
.
index whereas 
used above? What are the definitions of 
? that allow the integral to be converted to a sum?
instead of 
from (32.88). Guessing that (32.88), and (32.93 on pg 587) where intended to be negated like done earlier (for example in (32.57)).
instead of 
instead of 
instead of 
, but many formulas on these pages continue to use the 
should be 
. There are also some missing positional indicators in (35.105) and (35.106).
should be 
.
. It should be 
.
should be 
.
and spin down 
states, the positive energy solutions 
are found to be
are found to be
and spin down state 
are also used to find one specific set of states for the positive energy solutions
for both the negative energy states, but I’ve used 
here for the negative states for consistency with the covariant equation solutions). 
are identified as solutions to the covariant Dirac equations 
, 
, where 
as follows
and 
states. Spelled out explicitly, these states for the z-axis spin up case are
, 
, and 
. Let’s verify all these relationships.
has been used, so we have a zero as desired, and no non-relativistic approximations are required. We can show this generally too, without requiring the specifics of the z-axis spin up or down solutions. This is actually even easier. For the positive energy solutions 2.4 we have
has been used. For the negative energy solutions 2.5 we have
in spherical coordinates also makes sense
and 
for 
and 
respectively)
, and 
this gives
with respect to either the positive energy or negative energy solutions.
should provide a space-time split of the Dirac equation. Let’s verify this
vectors is easy to show, and we can do so without requiring any specific spin orientation
vectors
for which we have 
(rather than the strict orthonormality condition 
). Consider the calculation of the Fourier coefficients of a state
, so that the coefficient is
(the two by two identity), so that we are left with
) we get the block identity matrix as desired.
as 
![\begin{aligned}\left[{\Sigma},{\gamma^\nu}\right] = e_\mu^{.\nu} \gamma^\mu,\end{aligned} \hspace{\stretch{1}}(3.45)](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Cleft%5B%7B%5CSigma%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D+%3D+e_%5Cmu%5E%7B.%5Cnu%7D+%5Cgamma%5E%5Cmu%2C%5Cend%7Baligned%7D+%5Chspace%7B%5Cstretch%7B1%7D%7D%283.45%29&bg=fafcff&fg=2a2a2a&s=0 "\begin{aligned}\left[{\Sigma},{\gamma^\nu}\right] = e_\mu^{.\nu} \gamma^\mu,\end{aligned} \hspace{\stretch{1}}(3.45)")
![\begin{aligned}\left[{ \gamma^\alpha \gamma^\beta e_{\alpha \beta} },{\gamma^\nu}\right]&=e_{\alpha \beta} \left[{ \gamma^\alpha \gamma^\beta },{\gamma^\nu}\right] \\ &=e_{\alpha \beta} \left( \gamma^\alpha \gamma^\beta \gamma^\nu-\gamma^\nu \gamma^\alpha \gamma^\beta \right) \\ &=e_{\alpha \beta} \left( \left( {\gamma^\alpha \cdot \gamma^\beta }+\gamma^\alpha \wedge \gamma^\beta \right) \gamma^\nu-\gamma^\nu \left( {\gamma^\alpha \cdot \gamma^\beta }+\gamma^\alpha \wedge \gamma^\beta \right) \right) \\ &=e_{\alpha \beta} \left(\left(\gamma^\alpha \wedge \gamma^\beta \right)\gamma^\nu-\gamma^\nu \left(\gamma^\alpha \wedge \gamma^\beta \right)\right)\\ &=e_{\alpha \beta} \left(\left(\gamma^\alpha \wedge \gamma^\beta \right) \wedge \gamma^\nu-\gamma^\nu \wedge \left(\gamma^\alpha \wedge \gamma^\beta \right)\right)+e_{\alpha \beta} \left(\left(\gamma^\alpha \wedge \gamma^\beta \right) \cdot \gamma^\nu-\gamma^\nu \cdot \left(\gamma^\alpha \wedge \gamma^\beta \right)\right)\\ &=2 e_{\alpha \beta} \left(\gamma^\alpha \wedge \gamma^\beta \right) \cdot \gamma^\nu\\ &=2 e^{\alpha \beta} \left(\gamma_\alpha \wedge \gamma_\beta \right) \cdot \gamma^\nu\\ &=2 e^{\alpha \beta} \left(\gamma_\alpha \delta_\beta^{.\nu}-\gamma_\beta \delta_\alpha^{.\nu}\right)\\ &=4 e^{\alpha \nu} \gamma_\alpha \\ &=4 e_\alpha^{.\nu} \gamma^\alpha \\ \end{aligned}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Cleft%5B%7B+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+e_%7B%5Calpha+%5Cbeta%7D+%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%5B%7B+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D+%5C%5C+%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%5Cgamma%5E%5Cnu-%5Cgamma%5E%5Cnu+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%5Cright%29+%5C%5C+%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28+%5Cleft%28+%7B%5Cgamma%5E%5Calpha+%5Ccdot+%5Cgamma%5E%5Cbeta+%7D%2B%5Cgamma%5E%5Calpha+%5Cwedge+%5Cgamma%5E%5Cbeta+%5Cright%29+%5Cgamma%5E%5Cnu-%5Cgamma%5E%5Cnu+%5Cleft%28+%7B%5Cgamma%5E%5Calpha+%5Ccdot+%5Cgamma%5E%5Cbeta+%7D%2B%5Cgamma%5E%5Calpha+%5Cwedge+%5Cgamma%5E%5Cbeta+%5Cright%29+%5Cright%29+%5C%5C+%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28%5Cleft%28%5Cgamma%5E%5Calpha+%5Cwedge+%5Cgamma%5E%5Cbeta+%5Cright%29%5Cgamma%5E%5Cnu-%5Cgamma%5E%5Cnu+%5Cleft%28%5Cgamma%5E%5Calpha+%5Cwedge+%5Cgamma%5E%5Cbeta+%5Cright%29%5Cright%29%5C%5C+%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28%5Cleft%28%5Cgamma%5E%5Calpha+%5Cwedge+%5Cgamma%5E%5Cbeta+%5Cright%29+%5Cwedge+%5Cgamma%5E%5Cnu-%5Cgamma%5E%5Cnu+%5Cwedge+%5Cleft%28%5Cgamma%5E%5Calpha+%5Cwedge+%5Cgamma%5E%5Cbeta+%5Cright%29%5Cright%29%2Be_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28%5Cleft%28%5Cgamma%5E%5Calpha+%5Cwedge+%5Cgamma%5E%5Cbeta+%5Cright%29+%5Ccdot+%5Cgamma%5E%5Cnu-%5Cgamma%5E%5Cnu+%5Ccdot+%5Cleft%28%5Cgamma%5E%5Calpha+%5Cwedge+%5Cgamma%5E%5Cbeta+%5Cright%29%5Cright%29%5C%5C+%26%3D2+e_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28%5Cgamma%5E%5Calpha+%5Cwedge+%5Cgamma%5E%5Cbeta+%5Cright%29+%5Ccdot+%5Cgamma%5E%5Cnu%5C%5C+%26%3D2+e%5E%7B%5Calpha+%5Cbeta%7D+%5Cleft%28%5Cgamma_%5Calpha+%5Cwedge+%5Cgamma_%5Cbeta+%5Cright%29+%5Ccdot+%5Cgamma%5E%5Cnu%5C%5C+%26%3D2+e%5E%7B%5Calpha+%5Cbeta%7D+%5Cleft%28%5Cgamma_%5Calpha+%5Cdelta_%5Cbeta%5E%7B.%5Cnu%7D-%5Cgamma_%5Cbeta++%5Cdelta_%5Calpha%5E%7B.%5Cnu%7D%5Cright%29%5C%5C+%26%3D4+e%5E%7B%5Calpha+%5Cnu%7D+%5Cgamma_%5Calpha+%5C%5C+%26%3D4+e_%5Calpha%5E%7B.%5Cnu%7D+%5Cgamma%5E%5Calpha+%5C%5C+%5Cend%7Baligned%7D+&bg=fafcff&fg=2a2a2a&s=0 "\begin{aligned}\left[{ \gamma^\alpha \gamma^\beta e_{\alpha \beta} },{\gamma^\nu}\right]&=e_{\alpha \beta} \left[{ \gamma^\alpha \gamma^\beta },{\gamma^\nu}\right] \\ &=e_{\alpha \beta} \left( \gamma^\alpha \gamma^\beta \gamma^\nu-\gamma^\nu \gamma^\alpha \gamma^\beta \right) \\ &=e_{\alpha \beta} \left( \left( {\gamma^\alpha \cdot \gamma^\beta }+\gamma^\alpha \wedge \gamma^\beta \right) \gamma^\nu-\gamma^\nu \left( {\gamma^\alpha \cdot \gamma^\beta }+\gamma^\alpha \wedge \gamma^\beta \right) \right) \\ &=e_{\alpha \beta} \left(\left(\gamma^\alpha \wedge \gamma^\beta \right)\gamma^\nu-\gamma^\nu \left(\gamma^\alpha \wedge \gamma^\beta \right)\right)\\ &=e_{\alpha \beta} \left(\left(\gamma^\alpha \wedge \gamma^\beta \right) \wedge \gamma^\nu-\gamma^\nu \wedge \left(\gamma^\alpha \wedge \gamma^\beta \right)\right)+e_{\alpha \beta} \left(\left(\gamma^\alpha \wedge \gamma^\beta \right) \cdot \gamma^\nu-\gamma^\nu \cdot \left(\gamma^\alpha \wedge \gamma^\beta \right)\right)\\ &=2 e_{\alpha \beta} \left(\gamma^\alpha \wedge \gamma^\beta \right) \cdot \gamma^\nu\\ &=2 e^{\alpha \beta} \left(\gamma_\alpha \wedge \gamma_\beta \right) \cdot \gamma^\nu\\ &=2 e^{\alpha \beta} \left(\gamma_\alpha \delta_\beta^{.\nu}-\gamma_\beta \delta_\alpha^{.\nu}\right)\\ &=4 e^{\alpha \nu} \gamma_\alpha \\ &=4 e_\alpha^{.\nu} \gamma^\alpha \\ \end{aligned}")
![\begin{aligned}a \cdot b &= \frac{1}{{2}} (a b + b a) = \frac{1}{{2}} \left\{{a},{b}\right\} \\ a \wedge b &= \frac{1}{{2}} (a b - b a) = \frac{1}{{2}} \left[{a},{b}\right] \\ a b &= a \cdot b + a \wedge b = \frac{1}{{2}} ( \left\{{a},{b}\right\} + \left[{a},{b}\right] )\end{aligned} \hspace{\stretch{1}}(3.45)](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7Da+%5Ccdot+b+%26%3D+%5Cfrac%7B1%7D%7B%7B2%7D%7D+%28a+b+%2B+b+a%29+%3D+%5Cfrac%7B1%7D%7B%7B2%7D%7D+%5Cleft%5C%7B%7Ba%7D%2C%7Bb%7D%5Cright%5C%7D+%5C%5C+a+%5Cwedge+b+%26%3D+%5Cfrac%7B1%7D%7B%7B2%7D%7D+%28a+b+-+b+a%29+%3D+%5Cfrac%7B1%7D%7B%7B2%7D%7D+%5Cleft%5B%7Ba%7D%2C%7Bb%7D%5Cright%5D+%5C%5C+a+b+%26%3D+a+%5Ccdot+b+%2B+a+%5Cwedge+b+%3D+%5Cfrac%7B1%7D%7B%7B2%7D%7D+%28+%5Cleft%5C%7B%7Ba%7D%2C%7Bb%7D%5Cright%5C%7D+%2B+%5Cleft%5B%7Ba%7D%2C%7Bb%7D%5Cright%5D+%29%5Cend%7Baligned%7D+%5Chspace%7B%5Cstretch%7B1%7D%7D%283.45%29&bg=fafcff&fg=2a2a2a&s=0 "\begin{aligned}a \cdot b &= \frac{1}{{2}} (a b + b a) = \frac{1}{{2}} \left\{{a},{b}\right\} \\ a \wedge b &= \frac{1}{{2}} (a b - b a) = \frac{1}{{2}} \left[{a},{b}\right] \\ a b &= a \cdot b + a \wedge b = \frac{1}{{2}} ( \left\{{a},{b}\right\} + \left[{a},{b}\right] )\end{aligned} \hspace{\stretch{1}}(3.45)")
commutes with any 
to make the first cancellation. We can do so by noting![\begin{aligned}\left[{\gamma^\alpha \gamma^\beta + \gamma^\beta \gamma^\alpha},{\gamma^\nu}\right] &= \left[{ 2 g^{\alpha \beta} \mathbf{1}},{\gamma^\nu}\right] \\ &= 2 g^{\alpha \beta} \left[{\mathbf{1}},{\gamma^\nu}\right] \\ &= 0\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Cleft%5B%7B%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%2B+%5Cgamma%5E%5Cbeta+%5Cgamma%5E%5Calpha%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D+%26%3D+%5Cleft%5B%7B+2+g%5E%7B%5Calpha+%5Cbeta%7D+%5Cmathbf%7B1%7D%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D+%5C%5C+%26%3D+2+g%5E%7B%5Calpha+%5Cbeta%7D+%5Cleft%5B%7B%5Cmathbf%7B1%7D%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D+%5C%5C+%26%3D+0%5Cend%7Baligned%7D+&bg=fafcff&fg=2a2a2a&s=0 "\begin{aligned}\left[{\gamma^\alpha \gamma^\beta + \gamma^\beta \gamma^\alpha},{\gamma^\nu}\right] &= \left[{ 2 g^{\alpha \beta} \mathbf{1}},{\gamma^\nu}\right] \\ &= 2 g^{\alpha \beta} \left[{\mathbf{1}},{\gamma^\nu}\right] \\ &= 0\end{aligned}")
![\begin{aligned}\left[{ \gamma^\alpha \gamma^\beta e_{\alpha \beta} },{\gamma^\nu}\right]&=e_{\alpha \beta} \left[{ \gamma^\alpha \gamma^\beta },{\gamma^\nu}\right] \\ &=e_{\alpha \beta} \left( \gamma^\alpha \gamma^\beta \gamma^\nu-\gamma^\nu \gamma^\alpha \gamma^\beta \right) \\ &=\frac{1}{{2}} e_{\alpha \beta} \left( \left( \left\{{\gamma^\alpha},{\gamma^\beta}\right\}+\left[{\gamma^\alpha },{ \gamma^\beta }\right]\right) \gamma^\nu-\gamma^\nu \left( \left\{{\gamma^\alpha },{\gamma^\beta }\right\}+\left[{\gamma^\alpha },{\gamma^\beta }\right]\right) \right) \\ &=\frac{1}{{2}} e_{\alpha \beta} \left( \left[{\gamma^\alpha },{ \gamma^\beta }\right] \gamma^\nu-\gamma^\nu \left[{\gamma^\alpha },{ \gamma^\beta }\right] \right) \\ &=\frac{1}{{2}} e_{\alpha \beta} \left[{\left[{\gamma^\alpha },{ \gamma^\beta }\right] },{\gamma^\nu}\right] \\ &=\frac{1}{{2}} e_{\alpha \beta} \left[{\left[{\gamma^\alpha },{ \gamma^\beta }\right] },{\gamma^\nu}\right] \\ &=\frac{1}{{2}} e_{\alpha \beta} \left[{\gamma^\alpha \gamma^\beta -\gamma^\beta \gamma^\alpha },{\gamma^\nu}\right] \\ &=e_{\alpha \beta} \left[{\gamma^\alpha \gamma^\beta },{\gamma^\nu}\right] \\ &=e_{\alpha \beta} \left(\gamma^\alpha \gamma^\beta \gamma^\nu-\gamma^\nu \gamma^\alpha \gamma^\beta \right) \\ &=e_{\alpha \beta} \left(\gamma^\alpha ( 2 g^{\beta \nu} - \gamma^\nu \gamma^\beta )-\gamma^\nu \gamma^\alpha \gamma^\beta \right) \\ &=e_{\alpha \beta} \left(2 \gamma^\alpha g^{\beta \nu} - \gamma^\alpha \gamma^\nu \gamma^\beta -\gamma^\nu \gamma^\alpha \gamma^\beta \right) \\ &=2 e_{\alpha \beta} \left(\gamma^\alpha g^{\beta \nu} - g^{\alpha \nu} \gamma^\beta \right) \\ &=2 e_{\alpha \beta} \gamma^\alpha g^{\beta \nu} + 2 e_{\beta \alpha} g^{\alpha \nu} \gamma^\beta \\ &=2 e_{\alpha}^{. \nu} \gamma^\alpha + 2 e_{\beta}^{.\nu} \gamma^\beta \\ &=4 e_{\alpha}^{. \nu} \gamma^\alpha \end{aligned}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Cleft%5B%7B+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+e_%7B%5Calpha+%5Cbeta%7D+%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%5B%7B+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D+%5C%5C+%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%5Cgamma%5E%5Cnu-%5Cgamma%5E%5Cnu+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%5Cright%29+%5C%5C+%26%3D%5Cfrac%7B1%7D%7B%7B2%7D%7D+e_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28+%5Cleft%28+%5Cleft%5C%7B%7B%5Cgamma%5E%5Calpha%7D%2C%7B%5Cgamma%5E%5Cbeta%7D%5Cright%5C%7D%2B%5Cleft%5B%7B%5Cgamma%5E%5Calpha+%7D%2C%7B+%5Cgamma%5E%5Cbeta+%7D%5Cright%5D%5Cright%29+%5Cgamma%5E%5Cnu-%5Cgamma%5E%5Cnu+%5Cleft%28+%5Cleft%5C%7B%7B%5Cgamma%5E%5Calpha+%7D%2C%7B%5Cgamma%5E%5Cbeta+%7D%5Cright%5C%7D%2B%5Cleft%5B%7B%5Cgamma%5E%5Calpha+%7D%2C%7B%5Cgamma%5E%5Cbeta+%7D%5Cright%5D%5Cright%29+%5Cright%29+%5C%5C+%26%3D%5Cfrac%7B1%7D%7B%7B2%7D%7D+e_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28+%5Cleft%5B%7B%5Cgamma%5E%5Calpha+%7D%2C%7B+%5Cgamma%5E%5Cbeta+%7D%5Cright%5D+%5Cgamma%5E%5Cnu-%5Cgamma%5E%5Cnu+%5Cleft%5B%7B%5Cgamma%5E%5Calpha+%7D%2C%7B+%5Cgamma%5E%5Cbeta+%7D%5Cright%5D+%5Cright%29+%5C%5C+%26%3D%5Cfrac%7B1%7D%7B%7B2%7D%7D+e_%7B%5Calpha+%5Cbeta%7D+%5Cleft%5B%7B%5Cleft%5B%7B%5Cgamma%5E%5Calpha+%7D%2C%7B+%5Cgamma%5E%5Cbeta+%7D%5Cright%5D+%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D+%5C%5C+%26%3D%5Cfrac%7B1%7D%7B%7B2%7D%7D+e_%7B%5Calpha+%5Cbeta%7D+%5Cleft%5B%7B%5Cleft%5B%7B%5Cgamma%5E%5Calpha+%7D%2C%7B+%5Cgamma%5E%5Cbeta+%7D%5Cright%5D+%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D+%5C%5C+%26%3D%5Cfrac%7B1%7D%7B%7B2%7D%7D+e_%7B%5Calpha+%5Cbeta%7D+%5Cleft%5B%7B%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+-%5Cgamma%5E%5Cbeta+%5Cgamma%5E%5Calpha+%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D+%5C%5C+%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%5B%7B%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%7D%2C%7B%5Cgamma%5E%5Cnu%7D%5Cright%5D+%5C%5C+%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%5Cgamma%5E%5Cnu-%5Cgamma%5E%5Cnu+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%5Cright%29+%5C%5C+%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28%5Cgamma%5E%5Calpha+%28+2+g%5E%7B%5Cbeta+%5Cnu%7D+-+%5Cgamma%5E%5Cnu+%5Cgamma%5E%5Cbeta+%29-%5Cgamma%5E%5Cnu+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%5Cright%29+%5C%5C+%26%3De_%7B%5Calpha+%5Cbeta%7D+%5Cleft%282+%5Cgamma%5E%5Calpha+g%5E%7B%5Cbeta+%5Cnu%7D+-+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cnu+%5Cgamma%5E%5Cbeta+-%5Cgamma%5E%5Cnu+%5Cgamma%5E%5Calpha+%5Cgamma%5E%5Cbeta+%5Cright%29+%5C%5C+%26%3D2+e_%7B%5Calpha+%5Cbeta%7D+%5Cleft%28%5Cgamma%5E%5Calpha+g%5E%7B%5Cbeta+%5Cnu%7D+-+g%5E%7B%5Calpha+%5Cnu%7D+%5Cgamma%5E%5Cbeta+%5Cright%29+%5C%5C+%26%3D2+e_%7B%5Calpha+%5Cbeta%7D+%5Cgamma%5E%5Calpha+g%5E%7B%5Cbeta+%5Cnu%7D+%2B+2+e_%7B%5Cbeta+%5Calpha%7D+g%5E%7B%5Calpha+%5Cnu%7D+%5Cgamma%5E%5Cbeta++%5C%5C+%26%3D2+e_%7B%5Calpha%7D%5E%7B.+%5Cnu%7D+%5Cgamma%5E%5Calpha+%2B+2+e_%7B%5Cbeta%7D%5E%7B.%5Cnu%7D+%5Cgamma%5E%5Cbeta++%5C%5C+%26%3D4+e_%7B%5Calpha%7D%5E%7B.+%5Cnu%7D+%5Cgamma%5E%5Calpha+%5Cend%7Baligned%7D+&bg=fafcff&fg=2a2a2a&s=0 "\begin{aligned}\left[{ \gamma^\alpha \gamma^\beta e_{\alpha \beta} },{\gamma^\nu}\right]&=e_{\alpha \beta} \left[{ \gamma^\alpha \gamma^\beta },{\gamma^\nu}\right] \\ &=e_{\alpha \beta} \left( \gamma^\alpha \gamma^\beta \gamma^\nu-\gamma^\nu \gamma^\alpha \gamma^\beta \right) \\ &=\frac{1}{{2}} e_{\alpha \beta} \left( \left( \left\{{\gamma^\alpha},{\gamma^\beta}\right\}+\left[{\gamma^\alpha },{ \gamma^\beta }\right]\right) \gamma^\nu-\gamma^\nu \left( \left\{{\gamma^\alpha },{\gamma^\beta }\right\}+\left[{\gamma^\alpha },{\gamma^\beta }\right]\right) \right) \\ &=\frac{1}{{2}} e_{\alpha \beta} \left( \left[{\gamma^\alpha },{ \gamma^\beta }\right] \gamma^\nu-\gamma^\nu \left[{\gamma^\alpha },{ \gamma^\beta }\right] \right) \\ &=\frac{1}{{2}} e_{\alpha \beta} \left[{\left[{\gamma^\alpha },{ \gamma^\beta }\right] },{\gamma^\nu}\right] \\ &=\frac{1}{{2}} e_{\alpha \beta} \left[{\left[{\gamma^\alpha },{ \gamma^\beta }\right] },{\gamma^\nu}\right] \\ &=\frac{1}{{2}} e_{\alpha \beta} \left[{\gamma^\alpha \gamma^\beta -\gamma^\beta \gamma^\alpha },{\gamma^\nu}\right] \\ &=e_{\alpha \beta} \left[{\gamma^\alpha \gamma^\beta },{\gamma^\nu}\right] \\ &=e_{\alpha \beta} \left(\gamma^\alpha \gamma^\beta \gamma^\nu-\gamma^\nu \gamma^\alpha \gamma^\beta \right) \\ &=e_{\alpha \beta} \left(\gamma^\alpha ( 2 g^{\beta \nu} - \gamma^\nu \gamma^\beta )-\gamma^\nu \gamma^\alpha \gamma^\beta \right) \\ &=e_{\alpha \beta} \left(2 \gamma^\alpha g^{\beta \nu} - \gamma^\alpha \gamma^\nu \gamma^\beta -\gamma^\nu \gamma^\alpha \gamma^\beta \right) \\ &=2 e_{\alpha \beta} \left(\gamma^\alpha g^{\beta \nu} - g^{\alpha \nu} \gamma^\beta \right) \\ &=2 e_{\alpha \beta} \gamma^\alpha g^{\beta \nu} + 2 e_{\beta \alpha} g^{\alpha \nu} \gamma^\beta \\ &=2 e_{\alpha}^{. \nu} \gamma^\alpha + 2 e_{\beta}^{.\nu} \gamma^\beta \\ &=4 e_{\alpha}^{. \nu} \gamma^\alpha \end{aligned}")
