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)

The definition of $\gamma ^{5}\,$ uses $\epsilon _{0123}=+1\,$, which is consistent with most other uses of the epsilon tensor (as a volume form). This is apparently "Bjorken and Drell". The proof, however, uses $\epsilon ^{0123}=+1\,$, which is from Peskin & Schroeder. I'm not going to change it until some consensus on convention is reached.--99.199.20.98 (talk) 17:52, 14 March 2009 (UTC)

Proof of $\gamma ^{5}$ uses $\epsilon ^{0123}=+1\,$, proof of trace identity number 6 uses $\epsilon ^{0123}=-1\,$.

At least in this article, only one convention should be used. -- Deuded (talk) 17:58, 4 December 2015 (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)

The sign convention here apparently follows Peskin and Schroeder, but they warn that other books may use different sign conventions...
Can someone name a book that uses $\gamma ^{0}={\begin{bmatrix}0&-I\\-I&0\end{bmatrix}}$, ${\vec {\gamma }}={\begin{bmatrix}0&{\vec {\sigma }}\\-{\vec {\sigma }}&0\end{bmatrix}}$, $\gamma ^{5}={\begin{bmatrix}I&0\\0&-I\end{bmatrix}}$ ?
It might be worth mentioning in the text. --Immer in Bewegung (talk) 05:45, 8 September 2008 (UTC)

Kaku, QFT, uses the above convention. (Michio Kaku, Quantum Field Theory, ISBN 0-19-509158-2, appendix A)--Michael C. Price ^talk 10:16, 9 October 2009 (UTC)

Itzykson & Zuber appendix A. The article now covers both chiral conventions. 67.198.37.16 (talk) 05:23, 24 November 2020 (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)

Hold on, they are tensors, but invariant tensors. That is, they are vectors and they remain unchanged, like the kronecker delta and Levi-Civita symbol tensors. --Michael C. Price ^talk 10:19, 9 October 2009 (UTC)

That's not what a tensor means formally; a tensor is something that that transforms like a tensor. That being said, aren't they tensors? In that slashed quantities are Lorentz scalars (which comes up later in the article), which arguably is part of the point for their existence since Dirac was trying to factor the Klein-Gordon equation and it doesn't make a lot of sense to have an operator which is the sum of quantities of different tensorial rank. Linkhyrule5 (talk) 13:30, 9 April 2023 (UTC)

What gamma matrices are from an abstract point of view is really subtle, and trying to say whether or not they're tensors in some sense is not quite the correct language for describing them. And at the level of formality needed to talk about them, this definition of 'a tensor is something that transforms like a tensor' is not a good operating definition.

For QFT, we look at tensors satisfying certain transformation properties under the Lorentz group which make them representations of the Lorentz group. So I'll define a tensor to be a representation of the Lorentz group. However, for discussing the Dirac equation you also need spinors, which are not representations of the Lorentz group but of the spin group of the Lorentz group. Then gamma matrices are intertwiners of the Dirac spinor representation and the vector representation.

So actually it doesn't make sense to talk about how they transform under Lorentz transformations at all. What they do is convert Lorentz transformations on Dirac spinors to Lorentz transformations on vectors. Zephyr the west wind (talk) 15:49, 9 April 2023 (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, e¹, e², e³}, 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)

I pasted the material verbatim from the second section of Dirac algebra as an alternate representation and then recommended the latter should be deleted as the material is covered here. I do hope that this material is not considered controversial.If so, perhaps we could discuss it?Selfstudier (talk) 15:33, 13 November 2010 (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 physics 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 thisargument 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)

I may be missing the point of this discussion but the original question can be simply answered since the $\gamma ^{\mu }$ matrices are unitary (See Dirac_equation). The anti-commutation relation:

$\displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu }I$

where $\eta ^{\mu \nu }\,$ is the Minkowski metric and $\ I\,$ is the unit matrix, yields the relations: $\left(\gamma ^{\mu }\right)^{2}=\eta ^{\mu \mu }I$.

Multiplying this equation by $\left(\gamma ^{\mu }\right)^{\dagger }$ from the left we get $\left(\gamma ^{\mu }\right)^{\dagger }\gamma ^{\mu }\gamma ^{\mu }=\eta ^{\mu \mu }\left(\gamma ^{\mu }\right)^{\dagger }$.

Unitarity of $\gamma ^{\mu }$ means $\left(\gamma ^{\mu }\right)^{\dagger }\gamma ^{\mu }=I$.

So: $\gamma ^{\mu }=\eta ^{\mu \mu }\left(\gamma ^{\mu }\right)^{\dagger }$

Or: $\left(\gamma ^{\mu }\right)^{\dagger }=\eta ^{\mu \mu }\gamma ^{\mu }$.

The signs in these relations are dependent upon the metric signature.

I have used $()^{\dagger }$ to represent the conjugate transpose operation.

tumeda (talk) 19:54, 15 December 2008 (UTC)

But how does it follow that $\left(\gamma ^{\mu }\right)^{\dagger }=\gamma ^{0}\gamma ^{\mu }\gamma ^{0}$?

130.92.9.55 (talk) 09:30, 3 October 2011 (UTC)

The section Euclidean Dirac matrices needs more text for clarity. Xxanthippe (talk) 23:57, 3 November 2009 (UTC).

Row echelon form

This might be a silly question because I'm just a layman. Why are the matrices not in row echelon form? -Craig Pemberton 08:20, 26 December 2009 (UTC)

Because the place of the rows is important. Switching the rows will change the physics. you don't just put matrices in "row echelon form" for pleasure. Setreset (talk) 10:26, 26 December 2009 (UTC)

Dirac-Pauli representation

Doesn't it make sense to add a Dirac-Pauli representation, as it seem to be the default representation? See the book "The Dirac Equation" of Thaller on page35 for instance (ISBN 3540548831). — Preceding unsigned comment added by 93.104.36.169 (talk) 21:38, 20 September 2011 (UTC)

Real Clifford representation of Dirac algebra and matrices

The lead makes the claim that the Dirac algebra is a Clifford algebra, in this instance Cℓ_1,3(R). Though it is clear to me that this statement (or a very closely related statement) is valid, it is not clear to me that this particular claim is directly supported by references, or for that matter that it is necessarily exactly true. It is however clear that although use of Cℓ_1,3(C) is convenient, constraints are necessary, making the complex algebra the wrong algebra. For rendering the Dirac equation, Hestenes favoured Cℓ_1,3(R) with certain tricks (even though he showed that the form of the equation could be recovered in each reference frame by choosing frame-specific constants and adapting the transforms on the spinor). A far more natural algebra in this context may be Cℓ_3,1(R), where this "trick" appears to be unneeded. My point is that if the article is to claim that a specific algebra is (isomorphic to) the Dirac algebra, it will be necessary to back this up with references, or else to indicate the ambiguity between the non-isomorphic algebras Cℓ_1,3(R) and Cℓ_3,1(R). I understand that in the physics community there is no consensus on this, and the latter would thus be appropriate. The article Dirac algebra must also be updated accordingly; it currently claims that it is Cℓ_1,3(C). — Quondum^☏✎ 07:53, 23 January 2012 (UTC)

Lounesto, page 128, says of Cℓ_1,3(R) [i.e. the algebra with signature + - - -] that it is "generated as a real algebra by the Dirac gamma-matrices γ₀, γ₁, γ₂, γ₃", while making no such similar statement in his immediately preceding section on Cℓ_3,1(R). Indeed, one can see immediately that this is the sign convention by squaring the matrices, since γ₀² = I and γ₁² = γ₂² = γ₃² = - I (Lounesto sec 9.9, first eqn, p. 128).

Lounesto at chapter 13 goes on to discuss the effect of changing between a (+ - - -) metric and a (- + + +) metric, taking the view that this should be seen as switching between two non-isomorphic mathematical descriptions of the same physical object, rather than between two different physical objects. Jheald (talk) 11:40, 23 January 2012 (UTC)

Ooops. Thanks for fixing my stupid mistake with the factors of (1/2). Not thinking straight at all! Jheald (talk) 12:05, 23 January 2012 (UTC)

Heh. I've often enough done that sort of thing myself.

On my issue, your first quote is clearly correct, but it says nothing about a "Dirac algebra", unless you take that as the algebra generated by the gamma matrices. And I don't find the second quote particularly illuminating with respect to the algebra itself (it presumably only refers to specific classes of object), perhaps due to my own lack of understanding. The real question is, of course, whether the concept "Dirac algebra" is actually defined generally (i.e. is an encyclopedic concept), and if it is, whether it is the algebra naïvely defined as the algebra as generated from these matrices, or the algebra as defined by isomorphism with these algebra that applies in the Dirac equation. The former (Cℓ_1,3(R)) seems to have no applicability in this context unless "complexified" (or rather translated into the applicable algebra Cℓ_3,1(R)). My impression from Google browsing is that "Dirac algebra" generally is loosely used to mean "the algebra generated by the Dirac matrices, but with an additional i thrown in wherever appropriate", hence not actually Cℓ_1,3(R). I should, of course, get Lounesto's book; he seems to be one of the most solid references on such issues. — Quondum^☏✎ 12:47, 23 January 2012 (UTC)

As of right now, there is a Dirac algebra article, written by User:Jheald in 2007 and edited by User:Quondum in 2012, which seems to say all the correct things. 67.198.37.16 (talk) 05:37, 24 November 2020 (UTC)

Physical Structure

Hi

The equation $\displaystyle \gamma ^{\mu }=(\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3})=\gamma ^{0}e^{0}+\gamma ^{1}e^{1}+\gamma ^{2}e^{2}+\gamma ^{3}e^{3}$ appearing in the section "Physical Structure" is just nonsense. It's not "misleading" as is pointed out later, it's nonsense. Why not just write like this:

The 4-touple $(\displaystyle \gamma ^{\mu })=(\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3})$ is often loosely described as a 4-vector. But ...

One of the virtues of the Einstein notation is that nonsense equations tend to reveal themselves directly;) YohanN7 (talk) 10:32, 11 September 2012 (UTC)

So, I finally rewrote it (completely). The title "Physical structure" hints more than that only LT properties should go in. But this is what was there before, and this is what is there now, hopefully better presented. YohanN7 (talk) 18:49, 2 November 2013 (UTC)

The Dirac Algebra

Hi.

As is mentioned, the Gamma matrices are part of an (16-dimensional) algebra. As far as that algebra goes, this article is far from complete. There is an article on the Dirac algebra. I think that either that article should be beefed up, or this on should be extended. YohanN7 (talk) 10:47, 11 September 2012 (UTC)

From The fifth gamma matrix, γ⁵: "Although uses the letter gamma, it is not one of the gamma matrices. The number 5 is a relic of old notation in which γ⁰ was called γ⁴."

This is true (I guess), but it is also true that the name particularly appropriate. The set {γ⁰,γ¹,γ²,γ³,γ₅} provide a Clifford algebra in five spacetime dimensions. I'll add mention of that if nobody objects. YohanN7 (talk) 23:15, 24 February 2013 (UTC)

Could we have a page reference to Weinberg for this (I see it is given as a reference)? I'd like to check the claim, specifically about the case of metric signature (1,4). —Quondum 15:43, 7 March 2015 (UTC)

Yes us could. Page 218 in section 5.5. Also, books have indices you know. Use them. The case (1,4) is proved in our article (to the extent that proofs should be detailed in our articles). It is done by the mere display of the two equations right before the statement. YohanN7 (talk) 13:14, 8 March 2015 (UTC)

Thank you. The reference only gives the case of {γ⁰,γ¹,γ²,γ³,γ₅}. From Classification of Clifford algebras, Cl_4,1(R)≈M₄(C) and Cl_1,4(R)≈M₂(H)², and are thus nonisomorphic as algebras. The first case, M₄(C), clearly is the matrix representation given in the article, but 4×4 complex matrices clearly cannot represent the (1,4) case: if it could, the two signatures would yield isomorphic Clifford algebras. The squaring and anticommutation equations in the article evidently are not sufficient conditions to ensure the dimension of the algebra. I have removed the mention of the (1,4) case from the article. —Quondum 18:26, 8 March 2015 (UTC)

Nonsense. That section, by the way, has broken TeX since you were in there. I did supply a reference, though I think it is overkill to have two references for the bleeding obvious. You can also have a look in Higher-dimensional gamma matrices#Generic odd d = 2k + 1. I suggest you take this to the reference desk if you have further questions. YohanN7 (talk) 19:05, 8 March 2015 (UTC)

Quondum is right. A free algebra on symbols $\gamma ^{i}$ modulo the appropriate anticommutation relations is a Clifford algebra, but when the $\gamma ^{i}$ are matrices, showing that they satisfy the anticommutation relations isn't enough, as they might also be related in ways that Clifford algebra basis elements shouldn't be. And indeed the fifth element of the "basis" $\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3},i\gamma _{5}$ is equal to ± the product of the first four (depending on sign convention), so they generate the same algebra as just $\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}$.

Interestingly, Weinberg seems to make the same error as you: he says "the anticommutation relations $\gamma _{5}^{2}=1$ and $\{\gamma _{5},\gamma ^{\mu }\}=0$, together with $\{\gamma ^{\mu },\gamma ^{\nu }\}=2\eta ^{\mu \nu }$, show that $\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3},\gamma _{5}$ provide a Clifford algebra in five spacetime dimensions" (p. 218, equations substituted inline, emphasis mine). -- BenRG (talk) 04:19, 9 March 2015 (UTC)

It is somewhat soothing that Weinberg (supposedly) makes the same mistake as me. It is also noteworthy that you immediately jump to the conclusion that Weinberg is making mistakes. This is truly great scholarship. Greater than Weinberg's. On the linear dependence; it is supposed to be there. See page 216 in Weinberg or the new reference I added to the article. See also my reply at the reference desk. YohanN7 (talk) 13:04, 9 March 2015 (UTC)

There is the possibility that the definition of a Clifford algebra differs between physics and mathematics. Physicists tend to define it as a set of matrices satisfying a set of anticommutation relations. This is also the definition used in the present article. The difference (if any) is that this may not be the "freest" algebra satisfying the anticommutation relations. If this now happens to be the case, it settles the issue. YohanN7 (talk) 14:20, 9 March 2015 (UTC)

Well, it seems that maybe you're both right. The Clifford algebra is the freest-such, and matrix representations will in general have more relations than the algebra. The gamma-5 is distinctly part of Cl(1,3), it's not linearly independent, and therefore cannot be a part of Cl(1,4) which requires one more linearly-independent basis vector. But we are in luck: the complexified Clifford algebra in odd dimensions is two copies of the algebra of one less dimension. That is, its an $\oplus$ of two copies. Calling these the left and right copy, and noting that gamma-5 is already behaving in a chiral fashion, then it seems like its safe to "intentionally" re-use it as one more generator. Now, this is explicitly adding a matrix relationship, where there wasn't one before, (if one sticks to the "freest possible" formal definition). It is not clear to me what the physical meaning of this accidental/intentional extra relation is. It's an extra discrete symmetry, and those discrete symmetries have a way of, uhh, "doing strange things", sometimes unexpected things. Anyway, this appears to be a generic effect for even vs odd-dimensional Clifford algebras. 67.198.37.16 (talk) 05:53, 24 November 2020 (UTC)

Incomplete proof "trace of any product of an odd number of gamma matrices is zero"

So far have shown for two cases, 1 and 3. Need to extend to general odd number (if I could I would)... Hai2410 (talk) 00:18, 30 January 2013 (UTC)

Never mind i've sussed it Hai2410 (talk) 00:53, 30 January 2013 (UTC)

Miscelaneous property 5 (proof lacking)

Why there is no proof nor reference to source for misc property 5? — Preceding unsigned comment added by 186.137.53.184 (talk) 03:19, 20 September 2013 (UTC)

Why not write one yourself? Xxanthippe (talk) 04:43, 20 September 2013 (UTC).

I wrote one (I am the same "anonimous"), in spite I cannot use the formula editor properly. I hope someone will fix the style. Also, mu proof is not very elegant. — Preceding unsigned comment added by 186.59.60.106 (talk) 21:45, 21 September 2013 (UTC)

The lead

The following from the lead needs reformulation:

When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts.

It is true when taken with a grain of salt and in reverse order. It would be better to note that the generators of the Lorentz group has a spin representation expressible in terms of the gammas as presented in Gamma matrices#Physical structure. Applying this representation to an arbitrary 4-dimensional vector space yields a space of spinors. (The covariance/contravariance needs no mention, it will only add confusion.)

I'd appreciate if somebody could help me formulate this in good lead lingo. YohanN7 (talk) 14:02, 6 November 2013 (UTC)

Expressing the Dirac equation

To get from the Dirac equation to the KG equation one can operate simply with $(i\partial \!\!\!/-m)$ on the left. This is a simpler method than the one already described and I suggest changing the existing to this version. Why make things more complecated than they need to be. — Preceding unsigned comment added by 163.1.246.64 (talk) 15:20, 11 November 2013 (UTC)

The method you propose does not produce the Klein–Gordon equation:

$(i\partial \!\!\!/-m)(i\partial \!\!\!/-m)\psi =(-\partial \!\!\!/\partial \!\!\!/-im\partial \!\!\!/-im\partial \!\!\!/+m^{2})\psi =(-\partial \!\!\!/^{2}-2im\partial \!\!\!/+m^{2})\psi$

But the method presently in the article does:

$-(i\partial \!\!\!/+m)(i\partial \!\!\!/-m)\psi =(\partial \!\!\!/^{2}+m^{2})\psi$

If you have a reference that shows that you are correct, please give it. Even if you are correct, I fail to see why it would be simpler. — Quondum 16:17, 11 November 2013 (UTC)

Right.

But the whole section is rather silly. Why do we have to here "prove" (I'd use more quotation marks if my expressionistic standards (whatever that means) didn't prohibit me to) that the Dirac particles have mass m by appeal to the KG equation? I'll take the liberty of dumping this section where it belongs. YohanN7 (talk) 17:58, 11 November 2013 (UTC)

Cool. You're right: taking a step back to look at the bigger picture helps. — Quondum 19:28, 11 November 2013 (UTC)

You can get the KG equation as follows, consider the Dirac equation, $i\partial \!\!\!/\psi =m\psi$. You can substitute this into the equation you correctly derived, $(-\partial \!\!\!/\partial \!\!\!/-2im\partial \!\!\!/+m^{2})\psi =(-\partial \!\!\!/^{2}-m^{2})\psi =0$ or $(\Box +m^{2})\psi =0$. This is of course the same method as you use, however I was confused as the original said "operate with $-(i\partial \!\!\!/+m)$ on both sides", which I took to mean: $-(i\partial \!\!\!/+m)(i\partial \!\!\!/-m)(-)(i\partial \!\!\!/+m)$ as opposed to what you actually meant: $-(i\partial \!\!\!/+m)(i\partial \!\!\!/-m)\psi =-(i\partial \!\!\!/+m)0=...$.

I would also have to agree that perhaps this section has no place in this article, although it probably should be somewhere. — Preceding unsigned comment added by 129.67.144.39 (talk) 20:05, 11 November 2013 (UTC)

Perhaps in Klein–Gordon equation where it says "Furthermore, any solution to the Dirac equation (for a spin-one-half particle) is automatically a solution to the Klein–Gordon equation"? — Quondum 22:58, 11 November 2013 (UTC)

Yes, The KG-article should say (and prove) that any solution to any relativistic wave equation satisfies the KGE. YohanN7 (talk) 00:59, 12 November 2013 (UTC)

Ooo – isn't that pegging the bar a little high? (Or perhaps I should just ask what qualifies as a relativistic wave equation. Oh, and I suspect we would not normally call the Dirac equation a wave equation, so perhaps you meant any relativistic first-order differential equation?) — Quondum 01:06, 12 November 2013 (UTC)

No, it isn't pegging the bar very high at all. The relativistic wave equations are all wave equations that Lorentz symmetry (and QM in this context) allows and only those. YohanN7 (talk) 03:38, 12 November 2013 (UTC)

What am I missing? Lorentz symmetry does not imply that something is a solution to the KG equation, surely?

Roughly $ds^{2}=dt^{2}-d\mathbf {x} ^{2}\Rightarrow m^{2}=E^{2}-\mathbf {p} ^{2}$ in relativity. When the latter, which expresses 4-momentum conservation, invariance of 4-momentum squared under Lorentz transformations, is canonically quantized the Klein-Gordon equation follows in addition to other equations that may hold, like the Dirac, Weyl, Proca, and more generally, the Bargmann-Wigner equations. The result holds for the QFT versions as well. Special relativity is a pretty strong constraint. The exact form of the additional equations depends on under what representation of the Lorentz group the solutions are supposed to transform. There is one, or we wouldn't term the equation as "relativistic". (These additional equations can be expressed in different forms, but this isn't important here.) In this sense, the Klein-Gordon equation is the fundamental equation of relativity and quantum mechanics combined (for massive particles). (Note that the lhs of the equation above selects our Clifford algebra too. You knew that but somebody else might not know.) I don't know if this captures "all" in my original statement, but some lack of rigor is allowed on a talk page. (If it doesn't, I'll just go back and tweak the premises so that it does;). Edit: Here is one change in the premises: m is nonzero.)YohanN7 (talk) 13:03, 12 November 2013 (UTC)

One of the best references for this is Steven weinberg's "The Quantum Theory of Fields volume 1", chapter 5. YohanN7 (talk) 16:01, 12 November 2013 (UTC)

I'm sceptical. Too many "roughly"s, "canonical"s, and "representation"s. You make it sound as though the decades of work by Dirac and many others reached conclusions that are obvious implications of a single group of symmetries on spacetime: special relativity. And no, it doesn't select our Clifford algebra. Fundamental? No, merely a shared feature of several more fundamental equations. Beware of all the hidden axioms that you operate from. —Quondum 16:07, 12 November 2013 (UTC)

Quondum, you are trolling. Don't do that. Read Weinberg.

I'm not going rigorous on a talk page, except for these remarks: $ds^{2}=dt^{2}-d\mathbf {x} ^{2}$ represents a quadratic form on spacetime. It selects our Clifford algebra up to the signature of the metric. I know that you attribute to that signature some importance. Few people do that.

The correct energy momentum relation for a free particle is $m^{2}=E^{2}-\mathbf {p} ^{2}$ and nothing else. This is fundamental, and it is the KG equation when quantized. Which are the several more fundamental equations in this context than the energy-momentum relation? Please explain.

Yes, all follows from a group single group of symmetries, that of special relativity, and the principles of quantum mechanics. What the hell else What else would it (or anything in RQM) follow from? The weather in California?

And it all, by the way, is the result of decades of work by Dirac and many others(precisely as you put it). The most important figure in this context is Wigner.

Now, as for hidden axioms, you shouldn't write on my nose (again!) that I'm not rigorous. I'm not going rigorous here because it's a talk page and you are most certainly trolling.

I'm not going to apologize (again) for being semi-harsh in one of my many replies to you. You seem to automatically assume, not for the first time, and probably not for the last time, that everything you don't know yourself about is automatically false. That is a bad attitude. YohanN7 (talk) 17:28, 12 November 2013 (UTC)

Point taken. In my defence, you were suggesting an inclusion in an article. For the rest, none of this belongs on this page. —Quondum 19:18, 12 November 2013 (UTC)

No, you were the one suggesting inclusion in an article. YohanN7 (talk) 20:08, 12 November 2013 (UTC)

Antisymmetrization of $\gamma ^{5}$

Respected @Cuzkatzimhut: I am writing this in reference to your last comment "all points you might have in the talk page of that article first." (see: User_talk:P.Shiladitya#Revert_of_grammar_correction_in_Gamma_matrices_article). (Why I wrote on my talk page last time? Actually I prefer to write on user's talk page instead of talk page of a article as the second one more easily available to public.)

Kindly pardon me for disturbing you again (as I am damaging wiki and you have no interest in it).

First off all kindly pardon me as whatever I wrote in support of my edit, were completely wrong. However, I am still typing these for the following reason -

1. Your edit [8 February 2018] was like this - "$\delta _{\mu \nu \varrho \sigma }^{\alpha \beta \gamma \delta }$ is the type (4,4) generalized Kronecker delta in 4 dimensions, full antisymmetrization." (It is also the current state of the article - as you ordered me to keep your edit untouched.) I have a problem with the edit " full antisymmetrization".

2. You reverted my edit - [1], where I wrote "The method used here are called as antisymmetrization." You left the comment - " the ungrammatical and meaningless "The method used here is called as" was restored. There is no "method" in a definition. " on my talk page.

(a) [one] I am copying a line from the paragraph just before equation 8.13, page 14 - "Another set of techniques is the symmetrization and antisymmetrization of indices." Google shows that 'method' and 'technique' are synonyms.

(b) I think you heard the name of Sean M. Carroll. Could you please read page 199 of his world famous lecture [2]?

(c) What has been written on wikitonary - [3]? "The act......".

Next point is that you called $\delta _{\mu \nu \varrho \sigma }^{\alpha \beta \gamma \delta }$ as "Full antisymmetrization". You can not type "$\delta _{\mu \nu \varrho \sigma }^{\alpha \beta \gamma \delta }$ is antisymmetrization". Secondly, I also want to add that you can not use the word "full" for $\delta _{\mu \nu \varrho \sigma }^{\alpha \beta \gamma \delta }$. Why? Find yourself.

In conclusion, I want to say that anybody does not know everything. I respect you and thus I expect you to be more polite to a novice in future. And please do not force other to keep wrong incorrect edits in future.

Thanking you again, P.Shiladitya^✍talk 13:34, 2 March 2018 (UTC)

One of your links is inaccessible, and to the best of my understanding Carrol's p 199 is not telling me anything cogent or actionable here. I am sorry you felt addressed brusquely, but you have the distinct impression this is about you and who knows what, and not about provable clarity in the article. The article is not a tutorial. You have not convinced me yet you have read, internalized, and truly and fully accepted the definition of the fully antisymmetrized Kronecker symbol hyperlinked, that you feel strongly about. The original "full antisymmetrization", was in the way of a parenthetical verbless phrase qualifying the symbol. I now adduced "in" ahead of it, to possibly address your expressive anxiety about it. "Full" is absolutely crucial, as "all 4 of the indices are antisymmetrized, not some, yet again addressed in the hyperlinked definition to the symbol. I have to confess I missed most of the other points you appear to be striving to make here, however. This is not a forum. Cuzkatzimhut (talk) 16:10, 2 March 2018 (UTC)

Respected @Cuzkatzimhut:

   My statement -
       :$\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}=\gamma ^{[0}\gamma ^{1}\gamma ^{2}\gamma ^{3]}={\frac {1}{4!}}\delta _{\mu \nu \varrho \sigma }^{0123}\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\varrho }\gamma ^{\sigma }$
      This method is called antisymmetrization of $\gamma ^{5}$.

1. This is an operation and the name of operation is Antisymmetrization. - page 14 of [4] (if still the link does not work, please read 'The Poor Man's Introduction to Tensors' from [5]), page 19 of [6]; you didi not talk about this point, Now adding another new one

2. $\delta _{\mu \nu \varrho \sigma }^{\alpha \beta \gamma \delta }$ is called antisymmetrizer [7].

3. How can I call $\delta _{\mu \nu \varrho \sigma }^{\alpha \beta \gamma \delta }$ in "FULL" antisymmetrization when the defination of full antisymmetrization is here.

BTW why are you not providing any citation where it has been clearly mentioned that $\delta _{\mu \nu \varrho \sigma }^{\alpha \beta \gamma \delta }$ (without anything multiplied to it) is in full antisymmetrization form?

Thanking you again, P.Shiladitya^✍talk 18:50, 2 March 2018 (UTC)

For the life of me, I cannot see what is troubling you. Both of your external links fail to open for me. I have been providing this very wikilink for a month, as precisely hyperlinked in the article. Is it the normalization troubling you or the antisymmetry? The 4-4 tensor is a fully antisymmetric tensor, antisymmetric in both upper and lower indices, and thereby a full antisymmetrizer. Are you contesting or agreeing with the definitions, and, in either case, what more do you think is suitable for the article? It already details, explicitly, what is already defined in the WP hyperlink, namely full antisymmetry. Why do you have to say it yet again, essentially for the 4th time, using the comically defocussing term "method" ? Out of curiosity: are the blue hyperlinks visible/accessible to you, and don't you always click on them (or have them "hover" in inserts) as you are reading the article in your browser? I mean, none of this discussion belongs here, instead of the specific WP article for this very tensor. Cuzkatzimhut (talk) 19:51, 2 March 2018 (UTC)

Is $\gamma ^{0}$ wrong?

In all the Quantum Field Theory textbooks I've checked, including Schwartz's Quantum Field Theory and the Standard Model, Zee's Quantum Field Theory in a Nutshell, and Weinberg's Quantum Theory of Fields, it says that

$\gamma ^{0}=\left({\begin{matrix}0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{matrix}}\right)$

in the Dirac representation. But this does not match the value given at the beginning of this article! Is the value of $\gamma ^{0}$ wrong here?

Thanks, Gneisss (talk) 07:09, 30 October 2018 (UTC)

I figured it out -- all of these books were using the Weyl representation. Gneisss (talk) 23:51, 11 November 2018 (UTC)

The Dirac representation is nice, if one is learning about antiparticles for the first time, and nifty things like zitterbewegung. Once you settle in, the Dirac representation mostly just gets in your way, because you realize that left-right is what's important, not particle-antiparticle. It would seem that the newer books are apparently just dumping the older Dirac representation as not being very useful. Thus, the march of progress. I would add this to the article, except I can't figure out how to say it in an encyclopaedic tone. 67.198.37.16 (talk) 06:01, 24 November 2020 (UTC)

Squared mass

The article states the following about the Majorana representation: `The reason for making all gamma matrices imaginary is solely to obtain the particle physics metric (+, −, −, −) in which squared masses are positive'.

I do not find this to be very enlightening, since an observable like a squared mass is always positive. 2001:638:504:C07C:CDE7:3AF3:25C9:FC88 (talk) 11:21, 4 June 2019 (UTC)

A squared mass here should be taken to mean "the squared norm of a 4-momentum", which may indeed be negative depending on convention. Linkhyrule5 (talk) 13:28, 9 April 2023 (UTC)

Add a citation section

I would like to add a citation section. This makes it easier for the reader, a citation section for citations and a reference section for references. Is there any objection or concerns? TMM53 (talk) 06:28, 15 January 2024 (UTC)