Wikipedia:Reference desk/Archives/Mathematics/2015 March 8

= March 8 =

Solving equations
Honored wikipedians. I want to find a function p(r), for 0 ≤ r ≤ 1, and a constant B, such that
 * $$3 c \int _0^1 e^{B p(r)}r^2 dr=1$$

where c=0.49, and such that
 * $$p(r)={9 c^2 \over 4 \pi}\int_r^1 e^{B p(s)} s^{-2}\left(\int _0^s e^{B p(t)}t^2 dt\right)ds$$

I computed the following numerical solution iteratively r,.p,.B  0 0.0865575 18.0525 0.1 0.0856919 18.0525 0.2 0.0830952 18.0525 0.3 0.0787673 18.0525 0.4 0.0727083 18.0525 0.5 0.0649181 18.0525 0.6 0.0553968 18.0525 0.7 0.0441443 18.0525 0.8 0.0311607 18.0525 0.9 0.0164459 18.0525  1         0 18.0525 To my surprise it seems to satisfy $$p(r)=p(0)(1-r^2)$$ (r^2)+(%{.)p 1 1 1 1 1 1 1 1 1 1 1 Is this exact? Thank you. Bo Jacoby (talk) 03:59, 8 March 2015 (UTC).


 * Did you perhaps write the second equation incorrectly? I think that $$p(r)=p(0)(1-r^2)$$ is a solution to:


 * $$p(r)={9 c^2 \over 4 \pi}\int_r^1 e^{\pm B p(s)} s^{-2}\left(\int _0^s e^{\mp B p(t)}t^2 dt\right)ds$$


 * But not a solution to the equation you wrote where the exponentials have the same sign. Dragons flight (talk) 19:55, 8 March 2015 (UTC)

Thanks, Dragon flight! No, the exponentials should not have different signs. How did you manage to find that result? I tried WolframAlpha and got
 * integral_0^s e^(A^2 (1-t^2)) t^2 dt
 * = (e^(A^2) (sqrt(pi) erf(A s)-2 A s e^(-A^2 s^2)))/(4 A^3)

where $$A^2=B p(0)$$ and erf(A s) = $${2 \over \sqrt\pi}\int_0^{A s} e^{-t^2}dt$$, such that the result is
 * $$\int_0^s e^{A^2 (1-t^2)}t^2 dt ={\int_0^{A s} e^{A^2-t^2}dt -A s e^{A^2(1- s^2)}\over 2 A^3}$$

but I cannot do the second integration. Bo Jacoby (talk) 00:08, 9 March 2015 (UTC).


 * Upon taking a second look, I probably convinced myself of something that wasn't true. Anyway let's examine the second equation.  Which for simplicity I'll rewrite as:
 * $$p(r)=A \int_r^1 e^{p(s)} s^{-2}\left(\int _0^s e^{p(t)}t^2 dt\right)ds$$
 * Where I've absorbed B into p(0) and condensed the prefactor terms.
 * Now we can find the equivalent differential form via by taking two derivatives with respect to r.


 * $$p'(r)= -A e^{p(r)} r^{-2}\left(\int _0^r e^{p(t)}t^2 dt \right)$$
 * $$p''(r)= -A \left( \left(e^{p(r)} r^{-2}\right)'*\left(\int _0^r e^{p(t)}t^2 dt \right) + e^{p(r)} r^{-2} e^{p(r)}r^2 \right)$$
 * $$p''(r)= -A \left( \left(e^{p(r)} r^{-2}\right)'*\left(\int _0^r e^{p(t)}t^2 dt \right) + e^{2 p(r)} \right)$$


 * Substituting from the first derivative equation, this becomes:


 * $$p''(r)= -A \left( \left(e^{p(r)} r^{-2}\right)'*{ -p'(r) r^2 e^{-p(r)} \over A} + e^{2 p(r)} \right)$$
 * $$p''(r)= -A \left( e^{p(r)} \left(r^{-2} p'(r) + -2 r^{-3}\right)*{ -p'(r) r^2 e^{-p(r)} \over A} + e^{2 p(r)} \right)$$
 * $$p''(r) = p'(r)^2 + {-2 \over r} p'(r) - A e^{2 p(r)}$$


 * Which could be solved numerically, or by series expansion, etc. However, assuming I've done the math correctly, it definitely doesn't have a solution $$p(r)=p(0)(1-r^2)$$.  Dragons flight (talk) 03:16, 9 March 2015 (UTC)

Thanks again. Your initial simplification is brilliant. I'll return after further study. Bo Jacoby (talk) 17:38, 9 March 2015 (UTC). I made a huge error in the numerical calculation. The formula $$p(r)=p(0)(1-r^2)$$ is not at all valid. Thanks. Bo Jacoby (talk) 09:18, 13 March 2015 (UTC).

Solving for f(n) ≡ 0 mod p
For odd prime $$p$$ and polynomial $$f(n)$$ what are the more efficient ways to solve the following?

$$f(n) \equiv 0 \pmod p\,$$

Assume I have

$$f(n), \frac{d^mf}{dn^m} \forall m$$

and their modular multiplicative inverses available. (I am looking for the general case so I can apply this in case the function may become different). Currently I am doing linear searches which is Θ(p) which encounters problems when $$p$$ is large. Thieh (talk) 04:38, 8 March 2015 (UTC)
 * Is p so large as to make Berlekamp's algorithm inefficient? --catslash (talk) 15:21, 8 March 2015 (UTC). Apparently the Cantor–Zassenhaus algorithm is better (but I am not familiar with it). In either case, you will first need to split f(n) into square-free factors. --catslash (talk) 15:33, 8 March 2015 (UTC)
 * What about terms irreducible over integers? like $$n^2 + 13$$ or similar? Also the article seems to be talking about finding f(n) given p(n) instead of finding n given f and p. Thieh (talk) 19:07, 8 March 2015 (UTC)
 * I had assumed that n was restricted to being an integer mod p - but in that case you could just test each possible n from 0 to p-1 unless p was too large for this to be practical, and so perhaps p is also too large for these algorithms. The algorithms mentioned factor f(n) mod p into irreducible factors (over the integers mod p). If any of these factors is linear, then you have a solution (but perhaps you might as well just each possible n for this). Non-linear factors are the minimal polynomial (field theory)s for the non-integer-mod-p roots of f. --catslash (talk) 19:53, 8 March 2015 (UTC)
 * Factorization of polynomials over finite fields was what I was looking for. I was quite a bit noob in terminology and as such the way to apply the algorithm.  Thanks. Thieh (talk) 22:07, 8 March 2015 (UTC)

Proofs
Are there proofs on this

(-1)(-1)=1

0/x=0

0/0=?

x/0=? — Preceding unsigned comment added by 151.236.166.33 (talk) 18:02, 8 March 2015 (UTC)
 * 0/0 and x/0 are usually undefined in ordinary arithmetic. Also you can transform 0/x = 0 into 0 ⋅ 1/x which is covered by field axioms.  Thieh (talk) 19:11, 8 March 2015 (UTC)
 * Proving (-1)2=1. Let m=-1. Then by definition m satifies the equation m+1=0. Multiply both sides of this equation by m to get m2+m=0. Add one to get (m2+m)+1=1. Then m2+(m+1)=1. Substitute m+1=0 and m2+0 =m2 to get m2=1. Qvod Erat Demonstrandvm. Bo Jacoby (talk) 08:12, 9 March 2015 (UTC).

Expression
What is more correct expression in mathematic?

$$x+(-x)=0$$

or

$$-x+x=0$$

With the reason — Preceding unsigned comment added by 151.236.166.33 (talk) 18:15, 8 March 2015 (UTC)
 * Addition is commutative for most places (some exceptions exist) so I would see problems in neither. I would think it would be more appropriate to arrange it the same way as the line before.  Thieh (talk) 19:17, 8 March 2015 (UTC)
 * Even if addition is not commutative, it usually supports inverses. Whatever, both expressions are perfectly correct. Maproom (talk) 21:37, 10 March 2015 (UTC)

Gamma matrices
Per YohanN7's suggestion, I am raising this question here. I would appreciate holding back for a while, as this is a request for additional input from others.

The article Gamma matrices claims that (with suitable choice of the matrices) ${γ^{0},γ^{1},γ^{2},γ^{3},iγ^{5}} |undefined$ is the Clifford algebra of signature (1,4) and that ${γ^{0},γ^{1},γ^{2},γ^{3},γ^{5}} |undefined$ is the Clifford algebra of signature (4,1). Can $M_{4}(C)$ be used to represent both $Cl_{4,1}(R)$ and $Cl_{1,4}(R)$? Classification of Clifford algebras suggests that it is isomorphic for the signatures (4,1), (2,3) and (0,5) cases only. —Quondum 20:51, 8 March 2015 (UTC)


 * I think you're right. {γ0, γ1, γ2, γ3, iγ5} clearly isn't the basis for a Clifford algebra since the product of the five elements is 1. {γ0, γ1, γ2, γ3, γ5} clearly isn't the Clifford algebra of signature (4,1) since the elements square to +−−−+. I think {iγ0, iγ1, iγ2, iγ3, γ5} is a basis for the real Clifford algebra of signature (4,1), but don't quote me on that. -- BenRG (talk) 01:44, 9 March 2015 (UTC)


 * I can add a simple argument for the specific case in the article (though my intent with the question was to answer the more general matter of whether $M_{4}(C)$ can be used at all to represent $Cl_{1,4}(R)$): the basis elements that generate any real Clifford algebra, and all their products (without duplication within a product) form an R-linearly independent set. This condition can be shown to be violated by the first basis given above, if one uses the knowledge that $γ^{5} = iγ^{0}γ^{1}γ^{2}γ^{3}$, as defined in that article: the product of all five elements is $γ^{0}γ^{1}γ^{2}γ^{3}iγ^{5} = −iγ^{5}iγ^{5} = (γ^{5})^{2}$, which is a scalar, i.e. R-linearly dependent on first (empty) product. —Quondum 02:42, 9 March 2015 (UTC)


 * The algebra generated by $$\gamma_0,\gamma_1,\gamma_2,\gamma_3,\gamma_5$$ is Cl(2,3). This is isomorphic to $$M_4(\mathbb C)$$.  For the other one, I think there's an ambiguity in the notation $$i\gamma_5$$.  Does this mean i times $$\gamma_5$$ in the complex Clifford algebra generated by $$\gamma_0,\gamma_1,\gamma_2,\gamma_3$$? If so, the product $$\gamma_0\gamma_1\gamma_2\gamma_3(i\gamma_5)$$ would be equal to one, so they would naturally not be independent generators of a Clifford algebra, per BenRG. Or does $$i\gamma_5$$ instead mean $$i\otimes\gamma_5$$ in the complexification of the Clifford algebra on the five generators $$\gamma_0,\gamma_1,\gamma_2,\gamma_3,\gamma_5$$ (so $$\otimes$$ is the real tensor product)?  If we assume the latter, then we do get the algebra Cl(1,4).  This is $$M_2(\mathbb H)\oplus M_2(\mathbb H)$$.  The algebras $$M_4(\mathbb C)$$ and $$M_2(\mathbb H)\oplus M_2(\mathbb H)$$ are non-isomorphic (for instance, one is simple and the other is not).  I hope this clears things up.   Sławomir Biały  (talk) 14:23, 9 March 2015 (UTC)


 * It's neither of those. It's the real algebra generated by $$\gamma_0, \gamma_1, \gamma_2, \gamma_3, i\gamma_5 \in \mathbb M_4(\mathbb C)$$. That the matrix $$\gamma_5$$ has a name while $$i\gamma_5$$ doesn't has no effect on their use as generators. -- BenRG (talk) 19:18, 9 March 2015 (UTC)


 * You seem to believe that $$M_4(\mathbb C)$$ is not (isomorphic to) the complex Clifford algebra generated by $$\gamma_0,\gamma_1,\gamma_2,\gamma_3$$. That belief is wrong.  Otherwise you misread my post.   Sławomir Biały  (talk) 21:42, 9 March 2015 (UTC)

Perhaps the recursive construction of Higher-dimensional gamma matrices, Section 3.3, might shed the light you are seeking, if you simply restricted to d=2 and d=3. This is incontrovertibly standard stuff in physics practice in the trenches, and you simply "fail the course" if you don't stick to it, as you should.Cuzkatzimhut (talk) 14:32, 9 March 2015 (UTC)


 * @Sławomir: In the set ${γ^{0},γ^{1},γ^{2},γ^{3},γ^{5}} |undefined$ the first gammas are of course those appropriate to metric signature $(3, 1)$, not $(1, 3)$ (which was not communicated by Q), so your $(2, 3)$ conclusion is wrong. Everything relates to matrices, so multiplication by $i$ has just that meaning, multiplication entry-wise by $i$. Whether this takes you out of an algebraic structure is not an issue. The issue of concern is that one has a set of matrices satisfying
 * $$\displaystyle\{ \gamma^\mu, \gamma^\nu \} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu} I_D,$$
 * where the signature and dimension can be anything. This is the defining relation in mathematical physics (from what I can gather) and it is the definition in gamma matrices. It may be (seems more than likely) that the mathematical definition sometimes allows for "freer" (larger) algebras that satisfy the same relations. See section "Possibility" below. The definition of mathematical physics is naturally the appropriate one to use when discussing gamma matrices because this is what one is likely to find in the relevant literature. YohanN7 (talk) 17:43, 9 March 2015 (UTC)


 * Oops, I'm using the "wrong" signature convention, apparently. In your conventions gamma 0 through 5 would generate Cl(4,1), which is $$M_2(\mathbb H)\oplus M_2(\mathbb H)$$.  Also, the set $$\gamma_0,\gamma_1,\gamma_2,\gamma_3,i\otimes\gamma_5$$ generate Cl(3,2), which is $$M_4(\mathbb R)\oplus M_4(\mathbb R)$$.  This is not simple as an algebra, and has a quotient (replacing the $$i\otimes$$ by ordinary entrywise multiplication by i) that satisfies the same anticommutation relations.  So this gives a way to get the representations of Cl(3,2) from those of Cl(3,1).  Do I understand this correctly?  (Or are my signatures still wacko?)   Sławomir Biały  (talk) 18:25, 9 March 2015 (UTC)


 * In the second example, the (original) gammas are supposed to be the ones appropriate to the $(1,3)$ signature (sorry about the confusion, but I didn't write the OP). In this case, $iγ^{5}$ squares to $−I$ so, if anything, ${γ^{0},γ^{1},γ^{2},γ^{3},iγ^{5}} → {Γ^{0},Γ^{1},Γ^{2},Γ^{3},Γ^{4}} |undefined$ generates an algebra in which
 * $$\displaystyle\{ \Gamma^\mu, \Gamma^\nu \} = \Gamma^\mu \Gamma^\nu + \Gamma^\nu \Gamma^\mu = 2 \eta^{\mu \nu} I_5$$
 * holds for metric signature $(1, 4)$. Whether this is a Clifford algebra or not seems to depend on the definition of what constitutes a Clifford algebra. YohanN7 (talk) 18:41, 9 March 2015 (UTC)
 * The algebra is "degenerate" as per the Weinberg quote in the next section. YohanN7 (talk) 18:56, 9 March 2015 (UTC)
 * Ok, so it's more or less the same story in both cases. In your conventions, you only get half of what I would call the Clifford algebra.  Also, this algebra lacks what might be called a proper Z2 grading, and so it is just the even part of the full Clifford algebra.  These are the isomorphisms at Clifford algebra between the even algebras in one dimension higher.  Recovering the odd parts too then requires introducing some operator that anticommutes with all of the gammas, and commutes with $$\gamma_5$$.  (Something like "charge conjugation" I guess.)   Sławomir Biały  (talk) 19:09, 9 March 2015 (UTC)


 * Oh, it is not my conventions. (I have as many conventions as I have moral principles.) It just seemed like a nice day to wear a physicist hat. I'll change tomorrow Yes, it seems like the mystery is resolved. Thank you. If I understand the reference below correctly, the set ${γ^{0},γ^{1},γ^{2},γ^{3},−iγ^{5}} |undefined$ generates a different algebra from ${γ^{0},γ^{1},γ^{2},γ^{3},iγ^{5}} |undefined$ in the sense that they cannot be related by a similarity transformation. One follow up question would also be what the representation theory of $SO(D − 1, 1) ≈ SO(1, D − 1)$ looks like. What are the dimensions of the spin (projective) representations? But this is a separate issue. YohanN7 (talk) 19:52, 9 March 2015 (UTC)


 * You seem to be making this much more complicated than it really is. $$\gamma^5 = i \gamma^0\gamma^1\gamma^2\gamma^3$$ (modulo sign convention) and therefore the set $$\{\gamma^0,\gamma^1,\gamma^2,\gamma^3,i\gamma^5\}$$ is the same as the set $$\{\gamma^0,\gamma^1,\gamma^2,\gamma^3,-\gamma^0\gamma^1\gamma^2\gamma^3\}$$ whether you use it as a set of generators or anything else. Giving a name to $$\gamma^5$$ doesn't change its properties.
 * The generators $$\{\gamma^0,\gamma^1,\gamma^2,\gamma^3,\gamma^5\} = \{\gamma^0,\gamma^1,\gamma^2,\gamma^3,i\gamma^0\gamma^1\gamma^2\gamma^3\}$$ are effectively independent if you build a real algebra from them, since a factor of i distinguishes the fifth element from the product of the first four. The generators $$\{\gamma^0,\gamma^1,\gamma^2,\gamma^3,i\gamma^5\} = \{\gamma^0,\gamma^1,\gamma^2,\gamma^3,-\gamma^0\gamma^1\gamma^2\gamma^3\}$$ are not independent. You get the same algebra from them as from $$\{\gamma^0,\gamma^1,\gamma^2,\gamma^3\}$$. And you get the same algebra from $$\{\gamma^0,\gamma^1,\gamma^2,\gamma^3,-i\gamma^5\} = \{\gamma^0,\gamma^1,\gamma^2,\gamma^3,\gamma^0\gamma^1\gamma^2\gamma^3\}$$ -- BenRG (talk) 20:11, 9 March 2015 (UTC)


 * The naming conventions aren't mine. The gammas (0,1,2,3) in one case $(1,3)$ aren't the same as in the other case $(3,1)$ (see above). The generators $$\{\gamma^0,\gamma^1,\gamma^2,\gamma^3,\gamma^5\}$$ are not effectively independent according to Weinberg p. 216. YohanN7 (talk) 21:25, 9 March 2015 (UTC)
 * Well, that's not exactly what he says (I am tired), but I would be surprised if you get more real independent matrices out of $$\{\gamma^0,\gamma^1,\gamma^2,\gamma^3,\gamma^5\}$$ than $$\{\gamma^0,\gamma^1,\gamma^2,\gamma^3,i\gamma^5\}$$ (where the basic gammas differ in the two sets) because $i^{2} = − 1$.YohanN7 (talk) 21:50, 9 March 2015 (UTC)
 * Suppose the full (mathematical) Clifford algebra is generated by one set but not by the other. This would have implications for the dimensionality of the representations of $SO(1,4)$ and $SO(4,1)$. I don't think either of the isomorphic groups has a 4-dimensional representation (they have the same representation theory). (This is why I brought up the representation theory to begin with). YohanN7 (talk) 22:06, 9 March 2015 (UTC)


 * Let A = $$\left(\begin{smallmatrix} 1&0\\0&1 \end{smallmatrix}\right)$$ and let B = iA. The real algebra generated by {A,B} is larger than that generated by {A,iB}.
 * The differences between Weinberg and Wikipedia are not relevant here, but if that's not clear to you, we could also let A' = $$\left(\begin{smallmatrix} 0&-1\\1&0 \end{smallmatrix}\right)$$ and B' = −iA', for example. The definition of γ5 has a factor of i in it in both conventions.
 * If you want an example with anticommuting matrices, let A = $$\left(\begin{smallmatrix} 0&-1\\1&0 \end{smallmatrix}\right)$$, B = $$\left(\begin{smallmatrix} 0&1\\1&0 \end{smallmatrix}\right)$$, and C = iAB (or −iAB). Then A² = −1, B² = C² = 1, they pairwise anticommute, and {A,B,C} generates M2(C) while {A,B,iC}, {A,B,−iC}, and {A,B} all generate M2(R).
 * I don't really know representation theory, but Spin(p,q) and SO(p,q) are associated with the even subalgebra of Cℓp,q(R), which is always isomorphic to the even subalgebra of Cℓq,p(R). It's only when dealing with the full Clifford algebra or the Pin group that there's a difference between p+q and q+p dimensions. -- BenRG (talk) 07:50, 10 March 2015 (UTC)
 * If you say that $$\{\gamma^0,\gamma^1,\gamma^2,\gamma^3,\gamma^5\}$$ generate the full $32 = 2^{5}$ elements required by the mathematical version of Clifford algebra, then I take your word for it. Thank you. It makes sense since some set should according to the classification. If this is something that is agreed upon in full now, I'll sharpen the wording in the affected articles distinguishing clearly between the physicists definition and the mathematicians more restrictive definition of a Clifford algebra (in a couple of days time and if nobody gets there before me). YohanN7 (talk) 10:21, 10 March 2015 (UTC)

One way to see this is that Cl(4,1) is $$M_4(\mathbb C)$$ which is a simple algebra. So any non-trivial algebra of matrices that satisfies the anticommutation relations for Cl(4,1) must be isomorphic to it. Sławomir Biały (talk) 11:03, 10 March 2015 (UTC)

Reply
For an explicit statement of the disputed sentences in the article gamma matrices, please see and. I consider David Tong and Steven Weinberg being reliable references.

About dependencies in odd spacetime dimension: In spaces or spacetimes with with odd dimensionality, the totally antisymmetric tensors of rank $n$ can be linearly related by the conditions
 * $$\gamma^{[\mu_1}\gamma^{\mu_2}\cdots\gamma^{\mu_r]} \propto \epsilon^{\mu_1\mu_2\cdots\mu_d} \gamma_{[\mu_{r+1}}\gamma_{\mu_{r+2}}\cdots\gamma_{\mu_{d}]}, \quad r = 0, 1, \ldots, d-1,$$

where $ε$ is the totally antisymmetric symbol and the left hand side is taken as the identity matrix for $r = 0$. Under these conditions there are only $2^{d−1}$ independent tensors, requiring $γ$-matrices of dimensionality at least $2^{(d−1)/2}$.

So, yes, we can have dependencies. There is nothing in the definition of a Clifford algebra ruling it out.

About the representations in general: D even
 * $γ_{a} = γ_{a}^{&dagger;}$ ($a=1, ...,D$) are $2^{D/2}$×$2^{D/2}$ matrices, which are unique modulo a similarity transformation.
 * All $Γ_{a}$ are linearly independent.
 * All $2^{D/2}$×$2^{D/2}$ matrices can be decomposed into $γ_{a_{1}a_{1}...a_{n}}|undefined$ ($0 ≤ n ≤ D$).

D odd
 * $γ_{a} = γ_{a}^{&dagger;}$ ($a=1, ...,D$) are $2^{(D−1)/2}$×$2^{(D−1)/2}$ matrices, which are not unique; there are two representations.
 * Not all $Γ_{a}$ are linearly independent.
 * All $2^{(D−1)/2}$×$2^{(D−1)/2}$ matrices can be decomposed into $γ_{a_{1}a_{1}...a_{n}}|undefined$ ($0 ≤ n ≤ (D−1)/2$).

This wraps it up. Note the dependencies again. I don't know personally about the authors reliability, the reference is new to me, but someone I trust ensures me that it is good enough. You may also want to see Higher-dimensional gamma matrices that tells exactly the same story.


 * Appendix E
 * Appendix E

Possibility
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 disputed article. The difference (if any) is that this may not be the "freest" algebra satisfying the anticommutation relations. YohanN7 (talk) 14:09, 9 March 2015 (UTC)

The fact that there are two non-equivalent representations in odd spacetime dimension (physics convention) is incompatible with any of them being the Clifford algebra (mathematics convention). Could we have someone more chip in here? YohanN7 (talk) 14:53, 9 March 2015 (UTC)


 * That you can't answer these questions yourself suggests that you barely understand gamma matrices, which would ordinarily be fine for someone asking questions on the ref desk but is pretty disturbing for someone who's been editing the gamma-matrix article for years and who wrote and  yesterday. -- BenRG (talk) 19:18, 9 March 2015 (UTC)


 * Oh, that's how I learn - by writing Wikipedia articles about things I do not know. I'll never write about something I know completely, it would bore me to death. You might disapprove of the practice, but I can usually (as I did this time) back up my writings with references. I have never claimed to be an expert, whereas you immediately and incorrectly conclude "Weinberg is making mistakes". I believe there is a Nobel prize between you and him. He is known to make shockingly few mistakes in his publications.


 * Quondum and I have a long history of disagreement. This time none of us was wrong, just following different conventions, and I was following the convention of the article and of mathematical physics. This has not usually been the case before. I have had plenty of reason to have my alarm turned on whenever Q starts debating or reverting using words such as "surely" and "clearly" or "obviously". It often means he is about to say something wrong, see for instance the last thread on the same talk page. But, yes, I do regret posting at his page. YohanN7 (talk) 21:06, 9 March 2015 (UTC)