Commutation of gamma matrices
Mapsread
I can find all sorts of documentation regarding anti-commutation of
the gamma matrices, but almost nothing on commutation. I believe
commutation states [A,B] = AB-BA. I'm trying to determine the
commutation value for the gamma matrices when written as [gamma^u,
gamma^v] (actually, the tensors are mu and nu, but you know what I
mean). It occurs to me that I don't really know how to expand this
expression because there are no dummy indices.
If it did have dummy matrices, I would expect the commutation to be:
gamma^0*gamma^0 - gamma^0*gamma^0 + g^0*g^1 - g^1*g^0 ... + g^4*g^4 -
g^4*g^4.
At any rate, how do I expand [gamma^u, gamma^v]?
Thanks!
P.S. I found one cryptic reference that seems to imply the result is
the zero matrix. Is this true?
JEMebius
Not a clear-cut answer to your questions, but rather general advices and directions;
I write this reply in the wee small hours.
Main Wiki reference: http://en.wikipedia.org/wiki/Gamma_matrix
See http://en.wikipedia.org/wiki/Commutator#Anticommutator for the generally accepted
terminology.
Quotation from OP: "I believe commutation states [A,B] = AB-BA. "
Presumably you mean to say
"the =commutator= of A and B, notation: [A, B], is defined as AB - BA".
A heavy mix-up of terminology - no wonder you got astray.
Commutation is just reversion of the multiplication order. It is performed, like in group
theory, by conjugation; BA = A^-1 . AB . A, i.e. conjugate AB by A^-1.
The term "commutator" is just a word hinting at the discrepancy between AB and BA;
Perform two 3D rotations A and B about perpendicular axes over angles "a" and "b" in the
two possible orders.
You literally observe a discrepancy proportional to the product a.b of the rotation
angles, disregarding of course third- and higher-order terms.
Keyword: Campbell-Hausdorff formula.
Please sharpen your writing and your sense of research and make things as clear as
possible for yourself before even thinking of making a post in a newsgroup.
Remember: I am in a moody mood.
Side remark:
As regards studying gamma matrices and all that: please forget the Einstein summation
convention; write down all implied capital sigmas; write out =at least once in your
lifetime= completely all vectors, matrices and tensors as long as their numbers of
components do not exceed 4^4 = 256 (the full 4D, or rather the (3+1)D curvature tensor).
The Einstein convention was relevant for early 20th-century typesetters who did not have
sufficient sigma characters in stock to typeset books and articles containing tensor formulas.
Nowadays (AD 2009) the Einstein convention is still only relevant when delivering lectures
and taking lecture notes; time saving in writing down and copying Riemann-Christoffel
symbols and full curvature tensors may be 20% - 40%.
I guess many physicists and mathematicians show off their knowledge of all tings
Einsteinian by omitting summation sigmas from summation formulas wherever possible.
Happy studies: Johan E. Mebius
Mapsread
> Mapsread wrote:
> > Greetings,
>
> > I can find all sorts of documentation regarding anti-commutation of
> > the gamma matrices, but almost nothing on commutation. I believe
> > commutation states [A,B] = AB-BA. I'm trying to determine the
> > commutation value for the gamma matrices when written as [gamma^u,
> > gamma^v] (actually, the tensors are mu and nu, but you know what I
> > mean). It occurs to me that I don't really know how to expand this
> > expression because there are no dummy indices.
>
> > If it did have dummy matrices, I would expect the commutation to be:
> > gamma^0*gamma^0 - gamma^0*gamma^0 + g^0*g^1 - g^1*g^0 ... + g^4*g^4 -
> > g^4*g^4.
>
> > At any rate, how do I expand [gamma^u, gamma^v]?
>
> > Thanks!
>
> > P.S. I found one cryptic reference that seems to imply the result is
> > the zero matrix. Is this true?
>
> Not a clear-cut answer to your questions, but rather general advices and directions;
> I write this reply in the wee small hours.
>
> Main Wiki reference:http://en.wikipedia.org/wiki/Gamma_matrix
>
>
> - Show quoted text -
Greetings Mr. Mebius,
Thank you for your response. I apologize that I wasn't clear so I'll
try again.
I'm a hobbyist working through the Dirac equation and the derivation
of its general solution. At one point in the derivation, I see p-slash
times p-slash is p^2, where p is the four-momentum and p-slash is the
momentum in Feynman slash notation. My curiosity piqued at the absence
of the gamma matrices in the result.
I found a reference that derived a^2 from a-slash times a-slash, again
showing that the gamma matrices disappear (that is, equate to the
identity matrix). At one point in this derivation, it shows that
[gamma^mu, gamma^nu] = 0. Ultimately, this is my question: I'm trying
to figure out the reasoning behind this step. I suppose it's 0 because
mu and nu are the same, but it didn't appear that they were using that
fact (mu and nu being the same) at this point in the proof.
I thought I would just work through it myself and, embarrassingly, I
didn't know how to expand [gamma^mu, gamma^nu] because I didn't know
what was meant by mu and nu! That is, when I expand the expression, do
I replace mu and nu by sequences of 0 through 3 OR are the mu and nu
indices just simple placeholders for two numbers? That is, does the
expression imply the Einstein summation convention? (As opposed to
just two simple numbers?)
I'm not a student and only have books to learn from, which may offer
some explanation if my question still seems clumsy.
Regards,
JEMebius
Dear Mr Mapsread,
Like you I am not a theoretical physicist by training. So we are in a sense "brothers in
arms".
If you can tell me what textbooks you are studying then I can perhaps tune in as regards
the notational conventions (such as the signature: +--- versus -+++, or perhaps ---+;
+++-). I guess no two authors use completely identical conventions.
I myself have to recall all the nuts and bolts that come with the Dirac equation, so I
cannot yet give a satisfactory reply right now.
We could exchange regular EMail addresses and keep in touch outside the newsgroup. My
EMail address is jemebius at xs4all dot nl; " at " and " dot " are the usual well-known
characters.
Kind regards: Johan E. Mebius
Igor
> Greetings,
>
> I can find all sorts of documentation regarding anti-commutation of
> the gamma matrices, but almost nothing on commutation. I believe
> commutation states [A,B] = AB-BA. I'm trying to determine the
> commutation value for the gamma matrices when written as [gamma^u,
> gamma^v] (actually, the tensors are mu and nu, but you know what I
> mean). It occurs to me that I don't really know how to expand this
> expression because there are no dummy indices.
>
> If it did have dummy matrices, I would expect the commutation to be:
> gamma^0*gamma^0 - gamma^0*gamma^0 + g^0*g^1 - g^1*g^0 ... + g^4*g^4 -
> g^4*g^4.
>
> At any rate, how do I expand [gamma^u, gamma^v]?
I'll give you a hint. Multiply them out and compute the commutators
by hand. You'll find that they form a group.