# Leptons, Quarks and spin representation of LCTs

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Posted (edited)

Hi everyone, I would like to ask for opinions/expertises on the relations between quarks, leptons and spin representation of linear canonical transformations (LCTs) as described in this link https://en.m.wikipedia.org/wiki/Linear_canonical_transformation". Please find below a picture describing this relation as taken from the link.

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Edited by TheoM
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Posted (edited)

So what is your question ? The group is often useful for linear symmetry relations.

Edited by Mordred
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Posted (edited)
1 hour ago, Mordred said:

So what is your question ? The group is often useful for linear symmetry relations.

As shown in the picture, a classification of quarks and leptons is deduced . So my question is : Are this approach and the statements really correct ? Could you please give more explanations about it?  Thank you

Edited by TheoM
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Posted (edited)

ok well as far as the chart goes the details are correct

the main reason why the SM model uses the SU(n) groups is that tis group is compact which becomes important for renormalization as well as Feymann path integrals.

the SL(n) group is not compact which can lead to issues with renormalization even though both groups are closed groups

the isomorphism for the Lorentz/Poincare group for example is

$S0(3.1)\simeq Sl(2,\mathbb{C})/\mathbb{Z}_2$

spinors are defined to transform under the action of the $$SL2(\mathbb{C}$$ group.

So yes that link is accurate where it gets used as opposed to other group types depends on the state being described. However as shown here

$S0(3.1)\simeq Sl(2,\mathbb{C}/\mathbb){Z}_2$

you can have isomorphisms with other groups

the isomorphisms for SU(N) to the SL(2,c) group can be found here

which is what reference 6 of the link you gave which corresponds to the chart you posted employs

further details here

I should also forewarn you though that the Schrodinger equation of QM is not Lorentz invariant although the Dirac equations are. The operators used in QM (position and momentum)  are not employed in QFT (field and momentum). QM uses the Klein Gordon equations which are Lorentz invariant.

This is the purpose of reference 6

": Linear Canonical Transformations (LCTs) are known in signal processing and optics as the generalization of certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the canonical commutation relations characterizing the coordinates and momenta operators. In this work, the possibility of considering LCTs to be the elements of a symmetry group for relativistic quantum physics is studied using the principle of covariance"

reference 6 here

Edited by Mordred
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Posted (edited)

I find it a tad peculiar that the multiplets of the standard model are obtained[???] (correctly AFAICT) from this 3-parameter group that only acts on the frequency-time plane. Really? How does the E,t plane "know" about parity?

The multiplets displayed there are sure those of the standard model. And the fact that electrons are called "negatons" doesn't bother me too much. But where do these "boxes" come from? From this group whose only motivation is to generalise and unify Fourier and Laplace transforms, as well as other reparametrizations of the phase? I suppose Wick rotations can be found accommodation there too...

I don't know. I've been looking at the more readable page on LCT's,

And everything seems nice and dandy until we get to this table with the claim,

Quote

classification of quarks and leptons based on a five-dimensional theory and spin representation of linear canonical transformations

"a" theory? What theory?

This suspiciously sounds like a non-properly-curated addendum to the previous wikipedia article by some people intent on self-promotion.

I could be totally off-base, but I find the last paragraph of the wikipedia article very suspicious even though the table is that of the SM. They haven't shown to me to any degree of accuracy or plausibility that the table can be proven from the representations of the LCT.

Besides, the LCT is 3-parameter group. The standard model, OTOH, is what

SU(3)xSU(2)LxU(1) --> 8+3+1 internal

+ spin

Only counting the compact-group (quantised) degrees of freedom.

It doesn't add up in my mind. How did you come across this? What is you interest? Can you tell me more?

Edited by joigus
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Posted (edited)

Lol I think you became too used to Unitary and orthogonal groups. Joigus

Would it help to know SO(3.1) and SU(n) are both  subgroups of SL(2,c)/Z_2 ?

$sl(2,\mathbb{C})=su(2)\oplus isu(2)$

generators denoted e,f,h

[e,f=h]

[h,e]=2e

[h,f]=-2f

the 2C is the linear combination of e,f,h

$\pi (h)=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$

$\pi( e)=\begin{pmatrix}0&1\\0&0\end{pmatrix}$

$\pi h=\begin{pmatrix}0&0\\-1&0\end{pmatrix}$

$f_i,h_i,e_i$ i=1,2,3....r

however the set of complex cannot all commute so you need commutations

$[h_ih_j]=0$

$[h_i,e_j]=A_{ji}e_j$

$h_i,f_i]=-A_{ji}f_j$

$[e_i,f_j]=\delta_{ij}h_{ij}$

where $$A_{ij}$$ is the Cartan matrix ( I won't go through the ladder operators as they are fairly lengthy) however it can be expressed as

$[h_i,e_i]=\langle\alpha_j\rangle=\frac{2}{\langle\alpha_j,\alpha_j}\langle\alpha_j,\alpha_i\rangle_j=A_{ji}e_j$

$\begin{pmatrix}2&-1\\-1&2\end{pmatrix}$

the above is for SL(2C) for sl(3,C) the Cartan matrix is an 8 dimensional algebra of rank 2 which means it has a 2 dimensional Cartan sub algebra given as follows

$\pi(t_1)= \begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}$

$\pi(t_2)= \begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}$

$\pi(t_3)= \begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}$

$\pi(t_4)= \begin{pmatrix}0&0&1\\0&-1&0\\0&0&0\end{pmatrix}$

$\pi(t_5) =\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}$

$\pi(t_6)= \begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}$

$\pi(t_7)= \begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}$

$\pi(t_1)=\frac{1}{\sqrt{3}} \begin{pmatrix}0&1&0\\1&0&0\\0&0&-2\end{pmatrix}$

You may note the last is the Gell-Mann matrices

if we take the commutator between $$\pi(t_1)$$ and $$\pi(t_2)$$ we get $$[\pi(t_1),\pi(t_2)]=2i\pi(t_3)$$ which is familiar in the su(2) algebra. Thus we can define the following

$x_1=\frac{1}{2}t_1$

$x_2=\frac{1}{2}t_1$

$x_3=\frac{1}{2}t_3$

$y_4=\frac{1}{2}t_4$

$y_5=\frac{1}{2}t_5$

$z_6=\frac{1}{2}t_6$

$z_7=\frac{1}{2}t_7$

$z_8=\frac{1}{\sqrt{3}}t_8$

with change in basis

$e_1=x_1+ix_2$

$e_2=y_4+iy_5$

$e_3=z_6+iz_7$

$f_1=x_1+ix_2$

$f_2=y_4+iy_5$

$f_3=z_6+iz_7$

Now I should inform everyone that the basis and coordinates I am describing apply to Dynken diagrams and what I am describing apply to the root diagrams...

the basis above in matrix form is

$\pi(e_1)=\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}$

$\pi(e_2)=\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}$

$\pi(e_1)=\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}$

$\pi(f_1)=\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}$

$\pi(f_2)=\begin{pmatrix}0&0&0\\0&0&0\\1&0&0\end{pmatrix}$

$\pi(f_3)=\begin{pmatrix}0&0&0\\0&0&0\\0&1&0\end{pmatrix}$

$\pi(x_3)=\frac{1}{2}\begin{pmatrix}1&1&0\\0&-1&0\\0&0&0\end{pmatrix}$

$\pi(z_8)=\frac{1}{3}\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}$

@joigus That should help better understand the special linear group of Real as well as complex. Now knowing the above applies to Dynken diagrams will also help better understand the validity of the OPs link as well as the methodology.

@TheoM Hope this answers your question as well on the validity behind the LCT's and where they are applied in particle physics so yes the link overall you provided is valid

Edited by Mordred
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Posted (edited)

for reference

see 4.2 for rank 2 roots. which I supplied the relations above. The roots can be thought of as  2 dimensional vectors in a plane

some other helpful diagrams involving other groups as well.

this one includes the Coxeter diagrams (they act as symmetry reflections ) well explained in this link

hope that helps

Edit I too find the 5 dimensional part used in reference 6 of the OPs link a bit fishy going to look into that particular article in greater detail

Edited by Mordred
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Posted (edited)
11 hours ago, joigus said:

I find it a tad peculiar that the multiplets of the standard model are obtained[???] (correctly AFAICT) from this 3-parameter group that only acts on the frequency-time plane. Really? How does the E,t plane "know" about parity?

The multiplets displayed there are sure those of the standard model. And the fact that electrons are called "negatons" doesn't bother me too much. But where do these "boxes" come from? From this group whose only motivation is to generalise and unify Fourier and Laplace transforms, as well as other reparametrizations of the phase? I suppose Wick rotations can be found accommodation there too...

I don't know. I've been looking at the more readable page on LCT's,

And everything seems nice and dandy until we get to this table with the claim,

"a" theory? What theory?

I have the same questions

11 hours ago, joigus said:

It doesn't add up in my mind. How did you come across this? What is you interest? Can you tell me more?

I am interested in physics, particularly particle physics and the unification of quantum mechanics and general relativity. Someone told me about this. Thank you for your answer to the post.

5 hours ago, Mordred said:

for reference

see 4.2 for rank 2 roots. which I supplied the relations above. The roots can be thought of as  2 dimensional vectors in a plane

some other helpful diagrams involving other groups as well.

this one includes the Coxeter diagrams (they act as symmetry reflections ) well explained in this link

hope that helps

Edit I too find the 5 dimensional part used in reference 6 of the OPs link a bit fishy going to look into that particular article in greater detail

Thank you very much for the detailed answers and for the links. Any additional details from you will always be welcome.

Edited by TheoM
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Posted (edited)

Your welcome I wouldn't use the reference 6 it's not a method to learn particle physics from. While the LCTs and the SL(2C) group is part of particle physics the chart in reference 6 was generated using the SM Unitary groups.

The groups the SM model primarily uses is U(1), SU(2) and SU(3). Focus on those groups first as well as the Poincare group. SO(3.1).

The above is the groups for the SM model the LCTs while has uses that wiki page doesn't describe correctly where they are used in particle physics. Instead it's reference is a method in development

It is the Unitary groups mentioned above that you will find in any particle physics textbook and not what is described by the LCT wiki link.

Dynkin diagrams are advanced beyond the introductory level.

While the Unitary groups will provide the details for another representation (Feymann path integrals)

Edited by Mordred
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Posted (edited)

Thank you again.I would like to know more about it if it may help in the unification of quantum mechanics and gravity ?

How about the fact that it is written that Clifford algebra, spin representation of LCTs and quantum phase space are used ?

Edited by TheoM
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Wouldn't change anything as SU(n) is a subgroup of Sl(n,C) which is a subgroup of GL(N,C). The problem with renormalization for gravity isn't that we cannot renormalize for normal gravity ranges. You have two types of divergence. IR (infrared) ie divide by zero. This is easily fixed and already implemented. The other end of the spectrum is ultraviolet divergence. (We don't have any known limit to the mass term ) so the singularity condition of the BB and the singularity of a BH.

This is where the problem occurs. In math speak via QFT we can renormslize for 1 loop integrals but cannot renormalize for 2 loop liberals and higher.

The mathematical method used won't change this as the problem is where to set the upper limit.

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21 hours ago, Mordred said:

Lol I think you became too used to Unitary and orthogonal groups. Joigus

For a whole bunch of good reasons!

21 hours ago, Mordred said:

Would it help to know SO(3.1) and SU(n) are both  subgroups of SL(2,c)/Z_2 ?

Mm... Not really. I'm very familiar with $$SU(2)/\mathbb{Z}_{2}\cong SO(3)$$ and $$SL(2)/\mathbb{Z}_{2}\cong SO(3,1)$$ and their algebras, as with some non-unitary groups. But it is one thing to decompose groups into factors and quotients and do the analysis in terms of Dynkin diagrams, central charges, Casimir quadratics and what have you, and quite a different thing is to state where these groups become relevant and why. And what they help to break apart and study.

For example SU(2) can be said to embody how elementary particle "see" rotations in space (example: spin), or...

it can also be an internal symmetry group that tells us nothing about space and just refers to abstract directions in the Hilbert space (example: hypercharge).

It's still a mystery to me how parametric transformations in the "space" of energy and time (LCT's) is telling us anything at all about transformations in a completely different space (the space of colour, hypercharge, electric charge, and so on). I fail to see how even spin is included in the package.

And I'm not any the wiser now. But it's perfectly possible that I'm just too rusty on this, so my apologies in advance if that's the case.

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Posted (edited)
6 hours ago, joigus said:

For a whole bunch of good reasons!

Mm... Not really. I'm very familiar with SU(2)/Z2SO(3) and SL(2)/Z2SO(3,1) and their algebras, as with some non-unitary groups. But it is one thing to decompose groups into factors and quotients and do the analysis in terms of Dynkin diagrams, central charges, Casimir quadratics and what have you, and quite a different thing is to state where these groups become relevant and why. And what they help to break apart and study.

For example SU(2) can be said to embody how elementary particle "see" rotations in space (example: spin), or...

it can also be an internal symmetry group that tells us nothing about space and just refers to abstract directions in the Hilbert space (example: hypercharge).

It's still a mystery to me how parametric transformations in the "space" of energy and time (LCT's) is telling us anything at all about transformations in a completely different space (the space of colour, hypercharge, electric charge, and so on). I fail to see how even spin is included in the package.

And I'm not any the wiser now. But it's perfectly possible that I'm just too rusty on this, so my apologies in advance if that's the case.

Understood and I'm glad you recognize that these symmetry relations are internal symmetries and not spacetime symmetries.  To understand spin I would recommend taking time studying Cartan subalgebra. Here is the trick The synmmetry representations are expressed according to weights which correspond to eugenvalues. For example the quantum numbers of angular momentum all have their own weight under lie algebra however they also have their own weight diagram. (aka root diagram) for example

the spin j of a particle is given by

$U(\vec{\theta})=e\frac{i}{\hbar}\vec{\theta}\cdot \hat{J}$

where $$\hat{J}$$ is the three angular momentum operators whose representation will be given by $$2J+1$$ dimensional and $$\vec{\theta}$$ are the 3 parameters gives

$e^{\frac{i}{\hbar}\vec{\theta}\cdot J}|jm\rangle=\sum_{n}=-jC_nJn\rangle$

imposing

$U(\vec{\theta-1})U(\vec{\phi}(\theta_1\theta_2))$

results in subalgebra SU(2)

$[J_i,j_j]=i\hbar\epsilon_{i,j,k}J_k$

where raising and lowering operators are defined

$J_\pm=(J_1\pm iJ_2)/11/2$

there is the spin operations you mentioned.

for SU(3)

Now the Gell Mann matrices above has three basis states.

$|\Lambda\mu_1\rangle=\begin{pmatrix}1\\0\\0\end{pmatrix}$

$|\Lambda\mu_2\rangle=\begin{pmatrix}0\\1\\0\end{pmatrix}$

$|\Lambda\mu_2\rangle=\begin{pmatrix}0\\0\\1\end{pmatrix}$

where $${\mu_1,\mu_2,\mu_3}$$ are called two component weight vectors given by eugenvalues

$$H_1=\lambda_3/2$$ and $$H_2=\lambda_8/2$$ see Gell-Mann matrices above

$\Lambda \mu_1=(1/2,\sqrt{3}/6):|\Lambda \mu_2\rangle=(-1/2,\sqrt{3}/6):|\Lambda \mu_3\rangle=(0,-\sqrt{3}3/3)$

the above is your Dynkan spin representation of SU(3)

SU(3) has an eight dimensional root diagram which is an adjoint representation not shown above

for the OP understandably this will likely be over your head but also for other readers Group theory is a theory of representations these representations gives us tools to find and organize symmetry relations and antisymmetric relations. These representations have their own algebras (lie Algebra, Clifford algebra, Cartan algebra, etc,etc). They often use internal symmetries which can be thought of as (mathematical symmetries) though these can be also be physical quantities or probability quantities. In particle physics the state is a typically a probability wavefunction same for QFT. Lie algebra involves raising and lowering of Operators an operator has a requirement of being a minimal 1 quanta of action. (Langrangian)

Now we also have group symmetries homomorphism> a linear map between two lie algebras is homomorphic if it is non invertable.(useful for bosons aka symmetric) An isomophism is  invertable (fermions aka antisymmetric). Now lie algebras have subalgebras. Dynkin diagrams help us organize all the simple and semi-simple representations. In a sense it forms an atlas of our mappings. So SU(3) has 8 generators The Gell Mann matrices above. Each matric has its own root and hence its own weight that has its own weight diagram (aka root diagram which is a map). These maps can be a sub group of a larger group and vice versa. Dynkin diagrams also provide these details.

Edited by Mordred
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Posted (edited)

What then are the explicit links between your explanations and the classification that was given? and how about the part concerning what is called quantum phase space? I guess  it's not just spacetime but also includes momentum space? I may somehow understand your answers but I still not understand the relations between all these things  and the physics that is behind. Thank you

Edited by TheoM
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Posted (edited)

Here's what I mean:

According to Wikipedia, the definition of LCT's is,

$X_{(a,b,c,d)}(u)=\begin{cases} \sqrt{\frac{1}{ib}}\cdot e^{i\pi\frac{d}{b}u^{2}}\int_{-\infty}^{\infty}e^{-i2\pi\frac{1}{b}ut}e^{i\pi\frac{a}{b}t^{2}}x(t)\,dt, & \text{when }b\ne0,\\ \sqrt{d}\cdot e^{i\pi cdu^{2}}x(d\cdot u), & \text{when }b=0. \end{cases}$

So it's just an integral transform acting on the time-dependent part of the position-variable representation of the total wave function. IOW, this transformation does not depend on colour, electric charge, weak hypercharge, spin, or any of that. Nothing! It doesn't even touch those indices.

How can it provide classification into irreducible representations according to colour, electric charge, weak hypercharge, spin and all of that?

I don't see how it does, and I can't picture any way in which anybody can tell me how it does unless they have a theory, as promised in the wikipedia article:

Quote

classification of quarks and leptons based on a five-dimensional theory and spin representation of linear canonical transformations

What theory? Where is the theory? Does anyone have a theory to explain this utterly unbelievable statement that a group acting on one space helps classify objects defined in another (completely unrelated) space!!!?

I know about Cartan, and Gell-Mann matrices, and unitary representations of compact groups. I know all of that. But it doesn't even begin to address any of my concerns about this.

Edited by joigus
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On 6/4/2024 at 6:04 PM, Mordred said:

": Linear Canonical Transformations (LCTs) are known in signal processing and optics as the generalization of certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the canonical commutation relations characterizing the coordinates and momenta operators. In this work, the possibility of considering LCTs to be the elements of a symmetry group for relativistic quantum physics is studied using the principle of covariance"

reference 6 here

Sorry I missed this. There seems to be a correspondence between one and the other, right?

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I think your issue is not realizing that the linear translations are between groups. For example the last example I gave is translating between SU(2) to SU(3). The original groups contain the information your looking for. The LCTs is how to take that information from one group to another group.

13 minutes ago, joigus said:

Sorry I missed this. There seems to be a correspondence between one and the other, right?

Correct and what you are doing is using the correlations to establish how to transform from one group or map to another.

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OK. Then the Wikipedia article needs editing, because it's very confusing. It explains nothing of that. Not a thing.

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Posted (edited)

Agreed that wiki article is lousy on the correct details. It's almost as if someone who wrote it was half guessing what's involved. It's likely that it was written by someone who knows how an engineer uses it but doesn't understand how it's used in particle physics.

Edited by Mordred

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