[math-fun] The sign of the Minkowski metric --- a smoking gun at last?
For as long as I have known not very much about physics , experts have disagreed over the sign of the Minkowski metric: apparently particle physicists prefer +++- , relativists ---+ [Wikipedia]. An attempt to resolve the issue via geometric algebra, combined with an exotic experimental investigation into neutrinoless double beta decay ultimately foundered [Ber01]; passim. [ To conflate a metric with a Clifford algebra is in any case dubious, since mathematicians seem unable to agree on whether to associate +++- with Cl(3,1,0) or Cl(1,3,0) --- leading to confusion elsewhere in this area. ] My own less adventurous investigations encountered similar controversy over the signature for Clifford algebra tailored to 3-space Euclidean geometry: I unhesitatingly selected Cl(3,0,1) in order to retain positive magnitudes for all versors, whereas roboticists [Sel05] follow a preference of Ian Porteous for Cl(0,3,1) . [ This puzzled me until I realised that practical robotics can presumably find little application for an odd-grade versor. ] However, while contemplating the the computation of a versor X transforming one vector frame F into another G , I noticed what might just constitute a decisive practical distinction, at any rate between Cl(p,q) and Cl(q,p) --- when p (alone) is even, the sign of G = 1/X F X is switched by substituting for X its dual X~ !! [ Trivial example when p+q = 1 --- F = G = [e] , X = 1 , X~ = e ; in Cl(0, 1) we have 1/X F X = G , 1/X~ F X~ = -G ; whereas in Cl(1, 0) no versor transforms F to -G . ] At the moment any physical and geometrical significance of this is unclear to me. In particular, in Cl(p,q,r) when r > 0 duality is a much more delicate matter: indeed when all p,q,r > 0 , a consistent definition of the concept seems impossible to formulate. [ In this fashion I extricate myself from having to admit that perhaps Jon Selig was right after all. ] Can any physicists out there cast further light? Fred Lunnon https://en.wikipedia.org/wiki/Minkowski_space @book{[Sel05], author = {Jon.~M.~Selig}, title = {Geometric Fundamentals of Robotics}, publisher = {Springer}, year = 2005, comment = {ISBN 0-387-20874-7 ; includes Euclidean subspaces} } @article{[Ber01], author = {Marcus Berg and C\'ecile DeWitt-Morette and Shangjr Gwo and Eric Kramer}, title = {The Pin Groups in Physics --- C, P, and T}, journal = {Reviews in Mathematical Physics}, volume = {13}, year = 2001, pages = {953--1034}, note = {August, 2001; \url{arXiv:math-ph/0012006v1}},
It makes absolutely no difference which convention you choose. Particle physicists like +--- so that p^mu p_mu = E^2 - p^2 = m^2, rather than -m^2. Cosmologists are interested in the spatial part of the metric, and like it to be positive. There is no experiment possible that can distinguish these two choices. -- Gene From: Fred Lunnon <fred.lunnon@gmail.com> To: Geometric_Algebra <geometric_algebra@googlegroups.com>; math-fun <math-fun@mailman.xmission.com> Sent: Tuesday, June 21, 2016 9:42 AM Subject: [math-fun] The sign of the Minkowski metric --- a smoking gun at last? For as long as I have known not very much about physics , experts have disagreed over the sign of the Minkowski metric: apparently particle physicists prefer +++- , relativists ---+ [Wikipedia]. An attempt to resolve the issue via geometric algebra, combined with an exotic experimental investigation into neutrinoless double beta decay ultimately foundered [Ber01]; passim. [ To conflate a metric with a Clifford algebra is in any case dubious, since mathematicians seem unable to agree on whether to associate +++- with Cl(3,1,0) or Cl(1,3,0) --- leading to confusion elsewhere in this area. ] My own less adventurous investigations encountered similar controversy over the signature for Clifford algebra tailored to 3-space Euclidean geometry: I unhesitatingly selected Cl(3,0,1) in order to retain positive magnitudes for all versors, whereas roboticists [Sel05] follow a preference of Ian Porteous for Cl(0,3,1) . [ This puzzled me until I realised that practical robotics can presumably find little application for an odd-grade versor. ] However, while contemplating the the computation of a versor X transforming one vector frame F into another G , I noticed what might just constitute a decisive practical distinction, at any rate between Cl(p,q) and Cl(q,p) --- when p (alone) is even, the sign of G = 1/X F X is switched by substituting for X its dual X~ !! [ Trivial example when p+q = 1 --- F = G = [e] , X = 1 , X~ = e ; in Cl(0, 1) we have 1/X F X = G , 1/X~ F X~ = -G ; whereas in Cl(1, 0) no versor transforms F to -G . ] At the moment any physical and geometrical significance of this is unclear to me. In particular, in Cl(p,q,r) when r > 0 duality is a much more delicate matter: indeed when all p,q,r > 0 , a consistent definition of the concept seems impossible to formulate. [ In this fashion I extricate myself from having to admit that perhaps Jon Selig was right after all. ] Can any physicists out there cast further light? Fred Lunnon https://en.wikipedia.org/wiki/Minkowski_space @book{[Sel05], author = {Jon.~M.~Selig}, title = {Geometric Fundamentals of Robotics}, publisher = {Springer}, year = 2005, comment = {ISBN 0-387-20874-7 ; includes Euclidean subspaces} } @article{[Ber01], author = {Marcus Berg and C\'ecile DeWitt-Morette and Shangjr Gwo and Eric Kramer}, title = {The Pin Groups in Physics --- C, P, and T}, journal = {Reviews in Mathematical Physics}, volume = {13}, year = 2001, pages = {953--1034}, note = {August, 2001; \url{arXiv:math-ph/0012006v1}},
But the Pin groups of the two Clifford algebras are distinct. [Ber01] claim that detection of NDBD (which has inconveniently declined to attend the debate) would decide the issue. While I am not competent to pronounce on the validity of their (lengthy) argument, it has as far as I know not been challenged. WFL On 6/21/16, Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote:
It makes absolutely no difference which convention you choose. Particle physicists like +--- so that p^mu p_mu = E^2 - p^2 = m^2, rather than -m^2. Cosmologists are interested in the spatial part of the metric, and like it to be positive. There is no experiment possible that can distinguish these two choices.
-- Gene
From: Fred Lunnon <fred.lunnon@gmail.com> To: Geometric_Algebra <geometric_algebra@googlegroups.com>; math-fun <math-fun@mailman.xmission.com> Sent: Tuesday, June 21, 2016 9:42 AM Subject: [math-fun] The sign of the Minkowski metric --- a smoking gun at last?
For as long as I have known not very much about physics , experts have disagreed over the sign of the Minkowski metric: apparently particle physicists prefer +++- , relativists ---+ [Wikipedia]. An attempt to resolve the issue via geometric algebra, combined with an exotic experimental investigation into neutrinoless double beta decay ultimately foundered [Ber01]; passim.
[ To conflate a metric with a Clifford algebra is in any case dubious, since mathematicians seem unable to agree on whether to associate +++- with Cl(3,1,0) or Cl(1,3,0) --- leading to confusion elsewhere in this area. ]
My own less adventurous investigations encountered similar controversy over the signature for Clifford algebra tailored to 3-space Euclidean geometry: I unhesitatingly selected Cl(3,0,1) in order to retain positive magnitudes for all versors, whereas roboticists [Sel05] follow a preference of Ian Porteous for Cl(0,3,1) . [ This puzzled me until I realised that practical robotics can presumably find little application for an odd-grade versor. ]
However, while contemplating the the computation of a versor X transforming one vector frame F into another G , I noticed what might just constitute a decisive practical distinction, at any rate between Cl(p,q) and Cl(q,p) --- when p (alone) is even, the sign of G = 1/X F X is switched by substituting for X its dual X~ !!
[ Trivial example when p+q = 1 --- F = G = [e] , X = 1 , X~ = e ; in Cl(0, 1) we have 1/X F X = G , 1/X~ F X~ = -G ; whereas in Cl(1, 0) no versor transforms F to -G . ]
At the moment any physical and geometrical significance of this is unclear to me. In particular, in Cl(p,q,r) when r > 0 duality is a much more delicate matter: indeed when all p,q,r > 0 , a consistent definition of the concept seems impossible to formulate. [ In this fashion I extricate myself from having to admit that perhaps Jon Selig was right after all. ]
Can any physicists out there cast further light?
Fred Lunnon
https://en.wikipedia.org/wiki/Minkowski_space
@book{[Sel05], author = {Jon.~M.~Selig}, title = {Geometric Fundamentals of Robotics}, publisher = {Springer}, year = 2005, comment = {ISBN 0-387-20874-7 ; includes Euclidean subspaces} }
@article{[Ber01], author = {Marcus Berg and C\'ecile DeWitt-Morette and Shangjr Gwo and Eric Kramer}, title = {The Pin Groups in Physics --- C, P, and T}, journal = {Reviews in Mathematical Physics}, volume = {13}, year = 2001, pages = {953--1034}, note = {August, 2001; \url{arXiv:math-ph/0012006v1}},
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A somewhat related thing is the naming of the two square roots of -1 in the complex plane. I have long thought that the one in the upper half plane and named i (no reason either of these two things should change, of course) is actually the one that is in the lower half plane and named -i. And vice versa as well. Correcting these long-held misconception would cause profound changes to ripple through mathematics. —Dan
On Jun 21, 2016, at 9:42 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
For as long as I have known not very much about physics , experts have disagreed over the sign of the Minkowski metric:
Dan is too conservative. The real axis should be vertical, not horizontal. That way, the Galois symmetry between +i and -i would be reflected in the left-right symmetry of macroscopic phenomena in earth's gravitational well. This convention would work well on blackboards and whiteboards, but I imagine some students would interfere with this crusade for logical consistency by placing their textbooks face-up on a desk or table and reading from them in that position. We'd have to make that illegal, of course. :-) Jim Propp On Tuesday, June 21, 2016, Dan Asimov <asimov@msri.org> wrote:
A somewhat related thing is the naming of the two square roots of -1 in the complex plane. I have long thought that the one in the upper half plane and named i (no reason either of these two things should change, of course) is actually the one that is in the lower half plane and named -i. And vice versa as well.
Correcting these long-held misconception would cause profound changes to ripple through mathematics.
—Dan
On Jun 21, 2016, at 9:42 AM, Fred Lunnon <fred.lunnon@gmail.com <javascript:;>> wrote:
For as long as I have known not very much about physics , experts have disagreed over the sign of the Minkowski metric:
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Jim is right. I always wondered why -i was heavier. —Dan
On Jun 21, 2016, at 12:19 PM, James Propp <jamespropp@gmail.com> wrote:
Dan is too conservative. The real axis should be vertical, not horizontal. That way, the Galois symmetry between +i and -i would be reflected in the left-right symmetry of macroscopic phenomena in earth's gravitational well.
This convention would work well on blackboards and whiteboards, but I imagine some students would interfere with this crusade for logical consistency by placing their textbooks face-up on a desk or table and reading from them in that position. We'd have to make that illegal, of course.
:-)
Jim Propp
On Tuesday, June 21, 2016, Dan Asimov <asimov@msri.org> wrote:
A somewhat related thing is the naming of the two square roots of -1 in the complex plane. I have long thought that the one in the upper half plane and named i (no reason either of these two things should change, of course) is actually the one that is in the lower half plane and named -i. And vice versa as well.
Correcting these long-held misconceptions would cause profound changes to ripple through mathematics.
—Dan
On Jun 21, 2016, at 9:42 AM, Fred Lunnon <fred.lunnon@gmail.com <javascript:;>> wrote:
For as long as I have known not very much about physics , experts have disagreed over the sign of the Minkowski metric:
It's clear that this mistake was made even before complex numbers were invented, when somebody incorrectly concluded that the real number line was horizontal. Apparently this unsung genius thought that the difference between positive numbers and negative numbers was no more salient than the difference between left and right. On Tue, Jun 21, 2016 at 4:44 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Jim is right. I always wondered why -i was heavier.
—Dan
On Jun 21, 2016, at 12:19 PM, James Propp <jamespropp@gmail.com> wrote:
Dan is too conservative. The real axis should be vertical, not horizontal. That way, the Galois symmetry between +i and -i would be reflected in the left-right symmetry of macroscopic phenomena in earth's gravitational well.
This convention would work well on blackboards and whiteboards, but I imagine some students would interfere with this crusade for logical consistency by placing their textbooks face-up on a desk or table and reading from them in that position. We'd have to make that illegal, of course.
:-)
Jim Propp
On Tuesday, June 21, 2016, Dan Asimov <asimov@msri.org> wrote:
A somewhat related thing is the naming of the two square roots of -1 in the complex plane. I have long thought that the one in the upper half plane and named i (no reason either of these two things should change, of course) is actually the one that is in the lower half plane and named -i. And vice versa as well.
Correcting these long-held misconceptions would cause profound changes to ripple through mathematics.
—Dan
On Jun 21, 2016, at 9:42 AM, Fred Lunnon <fred.lunnon@gmail.com <javascript:;>> wrote:
For as long as I have known not very much about physics , experts have disagreed over the sign of the Minkowski metric:
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Maybe it's the extra printer's ink in the minus sign. Try renaming "i" as "+i" , and observe whether the diagrams trun trutle --- er, turn turtle. In the case of Clifford algebra, it's not the sign of a vector itself that signifies (so to speak), but the difference in signs, which --- in some fashion which I do not understand --- seems to be associated with electron spin: rotation of the vector through 2 pi changes its sign, although both vectors represent the same geometric reflection. WFL On 6/21/16, Dan Asimov <dasimov@earthlink.net> wrote:
Jim is right. I always wondered why -i was heavier.
—Dan
On Jun 21, 2016, at 12:19 PM, James Propp <jamespropp@gmail.com> wrote:
Dan is too conservative. The real axis should be vertical, not horizontal. That way, the Galois symmetry between +i and -i would be reflected in the left-right symmetry of macroscopic phenomena in earth's gravitational well.
This convention would work well on blackboards and whiteboards, but I imagine some students would interfere with this crusade for logical consistency by placing their textbooks face-up on a desk or table and reading from them in that position. We'd have to make that illegal, of course.
:-)
Jim Propp
On Tuesday, June 21, 2016, Dan Asimov <asimov@msri.org> wrote:
A somewhat related thing is the naming of the two square roots of -1 in the complex plane. I have long thought that the one in the upper half plane and named i (no reason either of these two things should change, of course) is actually the one that is in the lower half plane and named -i. And vice versa as well.
Correcting these long-held misconceptions would cause profound changes to ripple through mathematics.
—Dan
On Jun 21, 2016, at 9:42 AM, Fred Lunnon <fred.lunnon@gmail.com <javascript:;>> wrote:
For as long as I have known not very much about physics , experts have disagreed over the sign of the Minkowski metric:
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (6)
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Allan Wechsler -
Dan Asimov -
Dan Asimov -
Eugene Salamin -
Fred Lunnon -
James Propp