The symbol was supposed to be a boldface 1. Moreover, unordered HTML lists do not seem to work properly either. Is there any tool, here or elsewhere, to preview the rendering of a post?

]]>PART 4.

Let’s express the discussion from PART 3 differently.

We have two complex-linear maps between 3-dimensional unitary spaces; the target space is the same for both maps. We want to classify the possible pairs up to three separate unitary basis changes: on each domain and on the common target.

After introducing irrelevant bases everywhere, the task is to determine the orbit space of the following group action of the group on the set : For and ,

$$

Let be the identity matrix, and let . The group action of we just considered induces a group action of on . We want to determine its orbit space.

We expect the dimension of the orbit space to be the (real) dimension of minus the dimension of . That is, minus . I.e., .

Let’s consider the subset consisting of all such that and each have three distinct positive eigenvalues. This is an open dense subset of which is preserved by the action of . We expect the orbit space of the action on to be 10-dimensional as well.

Experiments show that the SM parameter lies in . Therefore physicists care only about the orbit space of .

The discussion from PART 3 boils down to the message that this orbit space can be parametrized by six numbers (the quark masses), which can be thought of as the noncompact part of the orbit space; and by the compact part, which can be thought of as the set of possible CKM matrices modulo a suitable group action. Namely the following action of the group on the set :

$$

The expected dimension of the compact part is the dimension of (i.e. ) minus the dimension of (i.e. ). Which yields an orbit space of expected dimension .

The whole point of this classification is that the three generation spaces are not identified but considered as separate spaces. This discussion can be found in physics textbooks (up to differences in notation and language, of course), for instance *O. Nachtmann: Elementary particle physics. Concepts and phenomena*, pp. 362–364.

We should continue with a translation of “flavor eigenstates” into the language above…

]]>The PMNS matrix for the leptons is analogous to the CKM matrix for the quarks, so let’s consider only the latter.

One thing that I've learnt over the course of the past few days (and discussed somewhat in the comments to Part 2 of these neutrino posts) is that flavour is defined differently for neutrinos than it is for quarks. For quarks and charged leptons, flavour states are mass states; for neutrinos, they're not. There are practical and historical reasons for this difference, but it is ultimately an arbitrary choice. So when you translate this discussion about quarks into a discussion about leptons, you turn u,c,b into e,μ,τ but d,s,t into ν₁,ν₂,ν₃. The PMNS matrix expresses νₑ etc as linear combinations of ν₁,ν₂,ν₃; analogously, the CMK matrix expresses d′,s′,t′ as linear combinations of d,s,t.

So the real distinction is not between mass states and flavour states but between mass states and weak states. For neutrinos, the flavour states are defined to be the weak states; for everything else, they're defined to be the mass states. The mass states, of course, are eigenstates of mass; the weak states are eigenstates of mass-after-emitting-or-absorbing-a-W-boson. That is, an interaction mediated by a W boson turns a mass state into a weak state or vice versa. It turns flavour states into flavour states if you’re talking about leptons, but not if you’re talking about quarks; this is only because the conventions are different, not the underlying mathematics and physics.

]]>Correction of the sentence with the “Formula does not parse” errors:

We just need sesquilinear forms and to write down the Yukawa terms.

]]>PART 3.

Now the Yukawa terms.

Let denote the Higgs field. There are two Yukawa quark terms. One involves , the other one involves . Only the generation tensor factors are relevant for our discussion. The Higgs field does not have a generation factor, hence can be ignored here.

We just need sesquilinear forms and to write down the Yukawa terms. Since is equipped with a (sesquilinear) scalar product (this is not the bilinear form from PART 2; clash of notation), we can identify with a -linear map via . Analogously corresponds to a -linear map .

These two maps are parameters of the SM Lagrangian and determine the Yukawa quark terms. (I will not describe these terms in detail. They involve invariant trilinear forms on the non-generation tensor factors. Again these trilinear forms are essentially unique.)

(The Yukawa terms are the only renormalizable two-fermions-one-Higgs Lagrangians which we can form from and which are allowed by general symmetry principles. The two maps parametrise all possibilities!)

Now we can figure out which decompositions of the vector spaces these maps induce.

Via the scalar products we get adjoints and . The obvious decompositions are:

the -eigenspace decomposition of (the corresponding eigenvalues are physically the squares of the down-quark masses);

the -eigenspace decomposition of (with the same eigenvalues as );

the -eigenspace decomposition of (the corresponding eigenvalues are physically the squares of the up-quark masses);

the -eigenspace decomposition of (with the same eigenvalues as ).

(If were somehow identified, we would get many additional endomorphisms and hence many other decompositions from the Lagrangian parameters !)

In the physically relevant (and mathematically generic) case, all eigenvalues are distinct. Hence (in this case) each decomposition is an orthogonal decomposition into three one-dimensional subspaces.

Now we have two orthogonal decompositions of into one-dimensional subspaces: one decomposition induced by , the other by . Let’s say is the eigenspace for the eigenvalue , where ; analogously with instead of .

If we fix in each of these subspaces a vector of norm 1, then we get a matrix describing the basis change from the -basis to the -basis. Each such matrix is called a "CKM matrix". There are many such matrices, because there are many ways to fix vectors of norm 1 in the eigenspaces: multiplying each of the six chosen vectors with an element of yields another possible choice. Each CKM matrix arising in this way describes the same physics: only the eigenspace decompositions are physically/operationally meaningful.

(to be continued some day…)

]]>[In PART 1, the numbering of my two-item ordered HTML list got lost in the translation.]

PART 2.

Let’s start with the first type of fermion-involving Lagrangian. (Is there a standard name for these terms, analogous to “Yukawa terms”?)

The (real-valued) bilinear form is determined by a sesquilinear form , as follows: is a section in the associated bundle constructed from the representation on $S_\psi\otimes V_\psi$, and is a section in another associated bundle constructed from a representation of on some tensor product . There is a nonzero sesquilinear form which is invariant under simultaneous application of on the left-hand resp.\ right-hand side. This sesquilinear form is unique up to multiplication with a complex number. We may pick any such form without loss of generality. From and we get in the obvious way a sesquilinear form whose real part gives us the bilinear form .

Without loss of generality we may assume that is hermitian: The antihermitian part yields a Lagrangian term that vanishes due to a partial integration argument (as long as we consider for instance compactly supported fields).

We can regard the hermitian forms as parameters of the SM Lagrangian. But, as an experimental fact, each of them is positive definite, i.e. a scalar product, for the universe we live in. (What does a universe look like where they have other signatures?)

Since all scalar products are isomorphic, they produce the same physics. In this sense, there is no need to include in the list of SM parameters. (Alternatively, we can include their isomorphism classes [i.e. their signatures] as a finite-valued parameter of the SM.)

To summarize PART 2: The a priori structureless generation vector spaces obtain scalar products from the first type of fermion-involving SM Lagrangians. They still have no decomposition into one-dimensional subspaces at this point, and there are no identifications between them.

(to be continued…)

]]>Apologies for the delayed reply. It’s still incomplete, and I see that Toby Bartels has by now said some things similar to what I wanted to say. Anyway, here is my take on the issue.

First one remark. You wrote: “Let me start by doing it operationally, as if I were an experimentalist. I’m sure you will find this infuriating […]” Quite the contrary. If someone doesn’t understand a physical concept, operational explanations are ultimately the only ones that can help. But I think we can figure this out without getting too operational.

You said: “If you succeed in doing the translation, please explain it to me!” I’ll try. I hope to avoid the old trap of “mathematicians translating something into their own language and henceforth it’s something completely different”.

PART 1.

Let . In the Standard Model (SM), fermionic fields of type correspond to sections in a certain complex vector bundle over spacetime, namely the associated vector bundle of some -principal bundle via a suitable representation of on a tensor product . Here is a representation of on a complex vector space ; and is the trivial representation of on a 3-dimensional complex vector space , which I’ll call the “generation space” of . In the following discussion we do not have to care too much about and . (You know what they are.)

The PMNS matrix for the leptons is analogous to the CKM matrix for the quarks, so let’s consider only the latter. The relevant fermion types are the (right-handed) up-quark field (in all three generations: up, charm, top), the (right-handed) down-quark field (in all three generations: down, strange, bottom), and the (left-handed) quark-field (a “doublet” in physics lingo).

The generation spaces are a priori not equipped with any structure: no scalar product, no decomposition into one-dimensional subspaces, no identifications between them. (You said that the three spaces should be identified. I hope to convince you that they shouldn’t.)

The (renormalizable part of the) SM Lagrangian contains two types of terms involving fermions:

Terms that describe interactions of two fermions (of the same type ) with a gauge boson. Schematically they look like , where is a certain first-order differential operator constructed from the gauge-boson field (a “twisted Dirac operator” in mathematical lingo). More about the bilinear form below.

Yukawa terms that describe interactions of two fermions (in our quark case: and either or ) and one Higgs boson.

(to be continued…)

]]>Wow, that’s cool! I’d seen DeWitt–Morette talk about how the Pin group works differently for (- + + +) and (+ – – -) signatures, and by now I know this math pretty well… but at the time I wasn’t at all interested in neutrinoless double beta decay, so I didn’t realize this is an *actual experiment* that could shed light on the signature of spacetime! I’ll have to reread this paper!

Nice pun, too.

]]>*“Anything anyone says about quantum gravity should be taken with a big grain of salt!”*

Such a grain should definitely be bigger than 0.00002 grams. :-)

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