Metric Signature Choice & Physical Consequences: Exploring Pin(1,3) & Pin(3,1)

[jean140]

TL;DR Summary

I want to know why the choice of a signature (-+++) or (+---) has no physical impact.

Hello,

I've always heard that the choice of signature for the metric was just a matter of convention, i.e. taking (+---) or (-+++) had no physical impact. The groups O(1,3) and O(3,1) being isomorphic it made sense to me.
However, I came across an article discussing the Pin(1,3) and Pin(3,1) non - isomorphic groups . It says, and I quote :

Two notable positive results show that the existence of two Pin groups is relevant to physics:

• In a neutrinoless double beta decay, the neutrino emitted and reabsorbed in the course of the interaction can only be described in terms of Pin(3,1).
• If a space is topologically nontrivial, the vacuum expectation values of Fermi currents defined on this space can be totally different when described in terms of Pin(1,3) and Pin(3,1).

So that would mean that it would be possible experimentally to find the true signature of the metric ? And so it would be wrong to say that the choice of signature is just a matter of convention ?

Thanks.

Edit : the article's link https://arxiv.org/abs/math-ph/0012006

 

[mfb]

It's just a different sign everywhere as long as you keep everything consistent. Imagine replacing all energies with their negative. You didn't change physical processes, you just flipped the sign of all energies.

 

[PeterDonis]

The groups O(1,3) and O(3,1) being isomorphic it made sense to me.

I don't think that's what the physical equivalence of the two metric signature conventions means. The choice of metric signature convention is a choice of sign for squared intervals: do you want timelike squared intervals to be positive and spacelike squared intervals negative (+---) or do you want timelike squared intervals to be negative and spacelike squared intervals to be positive (-+++). But either sign choice still has the same physical meaning: that you have timelike and spacelike squared intervals. Changing the signature convention does not change which intervals are timelike and which intervals are spacelike.

The question of what group of transformations on spacetime leaves physical quantities invariant is orthogonal to the signature convention choice.

 

[vanhees71]

Well, but the physics is indeed in the symmetry groups, which enable to construct the spacetime manifold (Minkowski space), and that's also how special relativity has been discovered by Poincare, Lorentz, and finally Einstein. On the other hand of course given the spacetime structure you can also find the symmetry group. It's a bit like the hen-and-egg problem. At the end it's irrelevant how you derive or assume the spacetime structure. That all the different sign conventions mean the same physics is indeed just the isomorphism between the manifolds and consequently the symmetry groups.

 

[jean140]

Hello and thank you all for your answers.

For my confusion about the Pin groups I understood that in reality you can fix the signature and simply change the definition of the generators of the group.

So that's it, everything is clear to me thank you.

 

[PeterDonis]

the physics is indeed in the symmetry groups

Both Pin groups are symmetry groups of Minkowski spacetime; the difference between them is which other things they do or do not leave invariant. So choosing one metric signature convention or the other doesn't change anything physically, although it might change how certain physical invariants are represented in the mathematics.

 

[vanhees71]

What are "Pin groups"? The symmetry group of SR is the proper orthochronous Poincare group. Some additional spacetime symmetries are approximately valid for electromagnetism and the strong interaction (space reflection, time reversal) but broken by the weak interaction.

 

[PeterDonis]

What are "Pin groups"

They're described in the article linked to in the OP.

 

[jean140]

What are "Pin groups"?

You can read p.18 - 20 here https://arxiv.org/abs/math-ph/0012006

 

[George Jones]

What are "Pin groups"? The symmetry group of SR is the proper orthochronous Poincare group. Some additional spacetime symmetries are approximately valid for electromagnetism and the strong interaction (space reflection, time reversal) but broken by the weak interaction.

I do not have time to read this long paper, but the abstract is fascinating.

Doesn't this say that nature could possibly favour one Pin group over the other?

Also, note that one of the authors is Cecile Dewitt-Morette.

Abstract A simple, but not widely known, mathematical fact concerning the coverings of the full Lorentz group sheds light on parity and time reversal transformations of fermions. Whereas there is, up to an isomorphism, only one Spin group which double covers the orientation preserving Lorentz group, there are two essentially different groups, called Pin groups, which cover the full Lorentz group. Pin(1,3) is to O(1,3) what Spin(1,3) is to SO(1,3). The existence of two Pin groups offers a classification of fermions based on their properties under space or time reversal finer than the classification based on their properties under orientation preserving Lorentz transformations — provided one can design experiments that distinguish the two types of fermions. Many promising experimental setups give, for one reason or another, identical results for both types of fermions. These negative results are reported here because they are instructive. Two notable positive results show that the existence oftwo Pin groups is relevant to physics:

•In a neutrinoless double beta decay, the neutrino emitted and reabsorbed in the course of the interaction can only be described in terms of Pin(3,1).

•If a space is topologically nontrivial, the vacuum expectation values of Fermi currents defined on this space can be totally different when described in terms of Pin(1,3) and Pin(3,1).

Possibly more important than the two above predictions, the Pin groups provide a simple framework for the study of fermions; it makes possible clear definitions of intrinsic parities and time reversal; it clarifies colloquial, but literally meaningless, statements. Given the difference between the Pin group and the Spin group it is useful to distinguish their representations, as groups of transformations on “pinors” and “spinors”, respectively.

The Pin(1,3) and Pin(3,1) fermions are twin-like particles whose behaviors differ only under space or time reversal.

 

[PeterDonis]

Doesn't this say that nature could possibly favour one Pin group over the other?

I think it is saying that there might be physical quantities associated with some particles (some kinds of fermions) that are only invariant under the transformations in one Pin group, as opposed to both.

I don't think it is saying that one metric signature convention might have a different physical meaning than the other.

 

[vanhees71]

I do not have time to read this long paper, but the abstract is fascinating.

Doesn't this say that nature could possibly favour one Pin group over the other?

Also, note that one of the authors is Cecile Dewitt-Morette.

A shorter version can be found in the book

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles,
Springer, Wien (2001).

 

[vanhees71]

I do not have time to read this long paper, but the abstract is fascinating.

Doesn't this say that nature could possibly favour one Pin group over the other?

Also, note that one of the authors is Cecile Dewitt-Morette.

The big question is, which Pin group Nature favors. I think it's related to the question, whether neutrinos are Majorana or Dirac fermions. This is not yet decided.

 

[jean140]

The big question is, which Pin group Nature favors. I think it's related to the question, whether neutrinos are Majorana or Dirac fermions. This is not yet decided.

What do you mean by "which Pin group Nature favors" ?
If I understood correctly it is possible that some particles are described by Pin(1,3) and others by Pin(3,1), and so Nature could choose one group or the other for differents particles (?).

 

[vanhees71]

Well, I've to read the paper in detail first, but as far as I understand, it's the question, how the discrete subgroups of the full Poincare group are realized in nature and how these discrete symmetries are broken. It makes a difference whether the neutrinos are Dirac or Majorana particles.