Talk:Gamma matrices

Identity 5

I'm not sure about trace identity 5, it seems that plugging in 0123 for \mu\nu\rho\sigma gives a sign mismatch. My field theory book has the identity with the sign as in the article, though, so I'm hesitant to change it.JochemKaas 01:33, 16 December 2006 (UTC) —The preceding unsigned comment was added by JochemKaas (talk • contribs) 01:37, 16 December 2006 (UTC)

Chiral definition

I fixed a sign on the chiral definition, as the text said that $\gamma ^{i}$ stays the same, as does my book... —The preceding unsigned comment was added by 131.111.225.229 (talk • contribs) 22:31, 12 January 2007 (UTC)

Not a vector

The gamma matrices, despite the suggestive notation, do not form a contravariant vector in any sense (neither a covariant vector or any other kind of vector). In fact, the gamma matrices remain unchanged under Lorentz transformations. A contravariant vector would change as v^{\prime \mu} = L^{\mu}_{\nu} v^{\nu}, where L is the Lorentz transformation matrix (and similar for covariant vectors).

...so, to be quite clear what I mean: The first sentence of the article is wrong, they don't form a 4-vector. —The preceding unsigned comment was added by 137.222.58.13 (talk • contribs) 13:17, 22 February 2007 (UTC)

Yes, and the section "Physical structure" goes into detail on exactly this issue. Please edit the lead to reflect that information! Melchoir 01:21, 23 February 2007 (UTC)

Missing Assumptions?

The first page of the following document www.damtp.cam.ac.uk/user/ho/SM.ps states that one ususally requires that gamma0 is hermitian and gamma1,gamma2,gamma3 are antihermitian. So IMHO the anticommutation relations are not enough to prove the identities in the "normalisation" paragraph. 88.103.65.231 17:02, 14 June 2007 (UTC)

The statement that these properties can be derived from the anticommutation-properties alone is wrong as the following example shows:

${\tilde {\gamma }}^{0}={\begin{pmatrix}I&-2I\\0&-I\end{pmatrix}},\quad {\tilde {\gamma }}^{i}={\begin{pmatrix}-\sigma ^{i}&2\sigma ^{i}\\-\sigma ^{i}&\sigma ^{i}\end{pmatrix}}$

By direct calculation or by noting that the ${\tilde {\gamma }}^{\mu }=A\gamma ^{\mu }A^{-1}$ where $\gamma ^{\mu }$ are the Dirac gamma-matrices and $A$ is given by

$A={\begin{pmatrix}I&I\\0&I\end{pmatrix}}$

one sees that the $\gamma ^{\mu }$ fulfill the same anticommutator-relations as the $\gamma ^{\mu }$ but of course they do not fulfill the claimed hermicity conditions

On a more abstract level this is of course to be expected, as the anticommutator-relations hold in any linear representation of the algebra of gamma-matrices on some vector-space $V$, so one can put an arbitrary scalar product on $V$ without changing the anticommutation-relations of the representatives of the $\gamma ^{\mu }$s on $V$. The hermiticity-property however refers to the scalar product on $V$, so it does depend on the specific choice of scalar product on $V$ and one can therefore not expect to derive it from the algebraic relations among the $\gamma ^{\mu }$s alone.

This shows that the hermicity-condition in fact is an additional requirement for a concrete realization (representation) of the gamma-matrices.

194.95.184.58 08:28, 30 October 2007 (UTC)

Expansion at the top

I dislike the "one possible representation [...] is" at the top, especially given that it doesn't seem to match any of the representations described below - it's the Dirac representation with a sign change for $\gamma ^{1}$ and $\gamma ^{3}$? I'd much rather have an explicit statement of the representation, but since I can't figure out why the signs are what they are, I don't want to edit...

Themel (talk) 21:05, 22 November 2007 (UTC)

It looks O.K. to me. The matrices are given in the Dirac basis, as at the end of the article. Xxanthippe (talk) 23:51, 22 November 2007 (UTC).

Nevermind. All sign errors are mine, as usual. Themel (talk) 07:16, 23 November 2007 (UTC)

Anons reverted

The edits made by the two anons since the beginning of February 2008, although undoubtedly done in good faith, do not improve the article. In my view, they obscure its meaning with attempts at spurious rigour, making the article less transparent. An explanation has been asked for but has not been forthcoming. Accordingly I have reverted to the version of 1 February. If I have reverted any useful edits, I apologise. Xxanthippe (talk) 23:40, 17 February 2008 (UTC).

Why I editted this page

Rather than adding spurious rigor I tried to explain

In mathematical physics, the gamma matrices, {γ0, γ1, γ2, γ3}, also known as the Dirac matrices, form a matrix-valued representation of a set of orthogonal basis vectors for contravariant vectors in space time, from which can be constructed a Clifford algebra.

As it stands I can see what is mend, but taken at face value it is rather mysterious. What is a matrix representation of a set of orthogonal basis vectors ????

Therefore I tried to explain that

1. the gamma matrices are a matrix version of something more fundamental: the representation of the abstract Clifford algebra Cl(V, g) generated by space time vectors V using the Minkowski metric g, on the complex space of spinors. The use of the mere symbol $gamma^{\mu }$ as opposed to the explicit matrix should be thought of as the physicists preferred notation for this representation.

2. like all matrix versions of a representations the explicit matrix entries of the gamma matrices depend on the choice of basis for V (and hence generators for Cl(V,g)), and a choice of basis for the spinors. This is obvious from the viewpoint of 1, and makes it obvious that the difference in choices by different authors really have about as much physical relevance as the difference in 19th century latitude longitude coordinates from English and French authors which used the null meridian through respectively the Greenwich or Paris observatory. This point seems to be widely misunderstood and responsible for a great amount of confusion on the nature of spinors and the Dirac equation. Understanding the Dirac equation in curved space time from the "the gamma matrices are fixed" point of view is even more confusing IMHO.

3. Given a set of 4 matrices {γ⁰, γ¹, γ², γ³}, and a basis {e⁰, γ¹, γ², γ³}, a necessary and sufficient condition for them to generate a representation of Cl(V,g) is that they satisfy the Clifford anti commutation relations.

4. Using a standard trick (the introduction of the Clifford group) one can see that the gamma matrices can be chosen to be unitary hence Hermtitian or skew-Hermitian.

5. One set of convenient conventions leads to the Dirac set, which is perfectly explicit and a good example.

R. Brussee —Preceding unsigned comment added by 195.169.16.163 (talk) 15:07, 27 February 2008 (UTC)

I have to say, I prefer the version that Xxanthippe (talk · contribs) has reverted to.

In mathematical physics, the gamma matrices, {γ0, γ1, γ2, γ3}, also known as the Dirac matrices, form a matrix-valued representation of a set of orthogonal basis vectors for contravariant vectors in space time, from which can be constructed a Clifford algebra.

It seems to me rather clear what this means. Given four orthogonal unit directions in space time, one might choose to represent each basis vector by a 4-tuple of numbers (1,0,0,0), (0,1,0,0), ... etc. But instead one can choose to represent each unit basis vector by a matrix, eg γ0, γ1, etc. This has the advantage that one can construct a Clifford algebra over these basis vectors, allowing one to represent unit bivectors, trivectors etc; and other goodies from geometric algebra, including Clifford representations of rotations, boosts etc.

Compare that to

In mathematical physics, the gamma matrices, {γ0, γ1, γ2, γ3}, also known as the Dirac matrices, are a set of complex 4 by 4 matrices that define an explicit matrix representation of the Clifford algebra constructed from contravariant space time vectors with their Minkowski metric.

This is immediately far less accessible. We can probably presume that almost all readers coming to this page have an idea what a basis vector is. The idea of using a matrix to numerically represent a unit basis vector may be a stretch for them, but it is an accessible one. But probably far fewer readers coming to this page know at the outset what a Clifford algebra is.

So it's far more appropriate for the lead to talk about "a matrix-valued representation of a set of orthogonal basis vectors, from which can be constructed a Clifford algebra;" instead of "an explicit matrix representation of the Clifford algebra constructed from contravariant space time vectors with their Minkowski metric."

The proposed text goes on:

The matrix representation of the Clifford algebra on C4 turns it into a spinor representation Δ, i.e. the irreducible complex representation of dimension 4 = 2^{4 / 2}. Since infinitesimal spatial rotations and Lorentz boosts can be represented as elements of the Clifford algebra, these also have representations on spinors. Spinors facilitate space-time computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-½ particles.

The Clifford algebra has only one spinor representation up to isomorphism. Therefore, all Lorentz invariant statements involving gamma matrices can be derived from their algebraic properties. From this perspective the gamma matrices are an index notation style way to speak about the abstract Clifford algebra together with a spinor representation. The symbol γμ is then more abstractly thought of as the linear operator on the representation space Δ defined by the space time basis vector eμ considered as an element of the Clifford algebra. A further choice of basis in the space of spinors Δ then determines a matrix for γμ whose explicit form depends on all the choices made. Index notations like {\gamma^{\mu\, A}}_B make all this manifest.

IMO this presupposes far too much knowledge for the WP:LEAD, and is likely to leave anyone not already substantially thinking along these lines completely bamboozled.

It may perhaps deserve a place further down the page, perhaps in a section called something like "A more abstract understanding of the gamma matrices.

As an aside, can I add that IMO it makes things far more complicated, to talk of particular elements of a Clifford algebra in terms of spinors, rather than talking of spinors in terms of particular elements of a Clifford algebra. The latter way round is IMO far more accessible. Jheald (talk) 18:47, 24 March 2008 (UTC)

Proof of adjoints

Can someone please insert a one line proof of how the anti-commutation relation implies

$\left(\gamma ^{0}\right)^{\dagger }=\gamma ^{0}\,$

and for the other gamma matrices (for k=1,2,3)

$\left(\gamma ^{k}\right)^{\dagger }=-\gamma ^{k}\,$

if indeed it is true without assuming specific forms of gamma matrices.Wiki me (talk) 16:33, 15 March 2008 (UTC)

Express the gamma matrices in terms of the alpha and beta matrices and note that the latter are self-adjoint. Xxanthippe (talk) 23:40, 15 March 2008 (UTC).

As I pointed out in reply to the comment following the heading "Missing Assumption", this statement (the anti-commutation relations imply the hermiticity condition) is in fact false; there I gave a counter-example to this statement and in addition provided an abstract argument why such a property cannot be expected to hold. Briefly, the point is that the concept of an (algebra-)representation does not necessarily refer to a scalar product; you can represent the Clifford algebra on a vector space that does not (yet) come equipped with a (natural) scalar product (think e.g. of the polynomials in one variable with degree less than or equal 3). Putting a scalar product on this vector space afterwards will of course not change the anticommutativity properties of the linear mappings representing your Clifford-Algebra elements, but it will change the question whether they are hermitian w.r.t this scalar product (in case you do not want to talk about hermiticity and adjoints in terms of some given scalar product but rather in terms of taking the transpose of a matrix and then conjugating each entry of the resulting matrix, the counterexample given in terms of concrete four by four matrices shows, that the more abstract argument indeed can be transfered to this concrete level as of course has to be case).

Concering the statement of the alpha and beta matrices, I guess what is meant here are the matrices appearing in the non-covariant formulation of the Dirac equation ("form originally proposed by Dirac" in the first section of the article "Dirac equation") ? However, the requirement in the "derivation" of the Dirac equation again is only the anticommutativity requirement, so again one _can_ choose the alpha and beta matrices in such a way that they are (anti-)hermitian, but one does not have to do that, so I do not see why this should be a proof for the claimed implication. 194.95.184.201 (talk) 08:54, 2 April 2008 (UTC)

In order to describe a physical observable an operator and its representations are required to be Hermitian. Xxanthippe (talk) 21:54, 2 April 2008 (UTC).

For operators representing physical observables, this is a perfectly legal argument (from a physicas point of view); for the

Dirac matrices however this does not apply directly because the observables in the dirac theory are build out of the bilinears (see e.g. the section "Dirac bilinears" at the end of the "Dirac equation" article), which (via the conjugate spinor) typically involve products of more than one dirac matrix. Requirering all matrices appearing in the definition of the bilinears to be hermitian now indeed gives restrictions; the ones following from this requirement for the scalar and the vector is the hermiticity of the zeroth gamma matrix and the hermiticity for the _products_ of the zeroth and i-th gamma-matrix. By using the anticommutation relations one sees, that the second requirement can be fulfilled by requirering _anti_-hermiticity for the i-th gamma-matrix whereas hermiticity of the gamma_i s is not compatible with the three requirements anticomm.-relns, herm. of gamma_0 gamma_i and herm. of gamma_0. This clearly shows that at least an application of the argument "operators representing physical observables have to be real" directly to the gamma-matrices is not possible (just because the dirac-matrices themselves do not represent physical observables). Applying this argument to the matrices appearing in the definition of the Dirac bilinears might well lead to the hermiticity conditions under discussion (I have not checked this, but it seems likely), but that would still not convince me that I have to use gamma-matrices fulfilling these conditions, because the spinor psi entering the Dirac bilinears in fact is not arbitrary but required to satisfy the Dirac equation where again the gamma matrices enter, so it might well be that this requirement on psi "conspires" with the specific form of the matrices appearing in the bilinears in such a way that the bilinears turn out real even if the matrices appearing in their definition are not hermitian (again this is just a guess).

Leaving all this reasoning aside (which of course is relevant from a physical point of view), the statement under discussion is a rather clear and purely mathematical one, namely "the anticommutation relations imply the (anti-)hermiticity conditions" and this statement is just false. There might be reasons to _impose_ this requirement (maybe physically motivated ones as the one discussed in the preceding paragraph, or just conventions), but this does not change the fact that this is an additional requirement, which does _not_ follow from the anticommutation relations. 194.95.184.201 (talk) 07:25, 3 April 2008 (UTC)