# Simple question on SUSY multiplets

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N=1 SUSY (massless) multiplets contain two particles with helicities differering by half a unit .

So one possible multiplet contains j=-1/2 and j=0. That corresponds to a left-handed spinor and a complex scalar so that's the chiral multiplet.

Another possibility is j=-1 and j=-1/2 which corresponds to a vector particle and a left-handed spinor (so a gauge multiplet).

I am confused by the fact that the helicity of a massless vector particle is -1. A photon, say, has two physical degrees of freedom. I know that helicity is not the same issue as teh issue of degrees of freedom since the left-handed spinor has two dof even if the helicity is -1/2. However, my confusion stems from the fact that a photon may be observed with right or left-handed polarization so it seemed to me that it has two possible helicity states. In which case it woul dnot make sense to assign only one helicity value to a gauge particle in a SUSY multiplet.

Clearly I am not understanding correctly the concept of helicity in the case of gauge particles. Can someone clear this up?

Thanks

blechman
N=1 SUSY (massless) multiplets contain two particles with helicities differering by half a unit .

So one possible multiplet contains j=-1/2 and j=0. That corresponds to a left-handed spinor and a complex scalar so that's the chiral multiplet.

actually, that's HALF of the chiral multiplet - the left handed spinor and a *REAL* scalar. Adding the CP-conjugate multiplet gives you j=+1/2 and j=0, the right handed spinor and another real scalar. So the two put together make the chiral multiplet.

Another possibility is j=-1 and j=-1/2 which corresponds to a vector particle and a left-handed spinor (so a gauge multiplet).

again, that's only HALF of the multiplet. Add the CP-conjugate and you've got the complete vector multiplet.

I am confused by the fact that the helicity of a massless vector particle is -1. A photon, say, has two physical degrees of freedom. I know that helicity is not the same issue as teh issue of degrees of freedom since the left-handed spinor has two dof even if the helicity is -1/2. However, my confusion stems from the fact that a photon may be observed with right or left-handed polarization so it seemed to me that it has two possible helicity states. In which case it woul dnot make sense to assign only one helicity value to a gauge particle in a SUSY multiplet.

Clearly I am not understanding correctly the concept of helicity in the case of gauge particles. Can someone clear this up?

Thanks

Cheers!
-A

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actually, that's HALF of the chiral multiplet - the left handed spinor and a *REAL* scalar. Adding the CP-conjugate multiplet gives you j=+1/2 and j=0, the right handed spinor and another real scalar. So the two put together make the chiral multiplet.
Thank you very much for replying. I feel like this is very helpful but I am still confused a bit.
First, to make sure we are using the same language, I am thinking in terms of Weyl spinors, not Majorana. So the left-handed spinor has two degrees of freedom, matched by the two degrees of freedom of the compelx scalar field.

A second related question: the two degrees of freedom of a left-handed Weyl spinor have j=-1/2 and j=+1/2 ? Even though they are both part of a left-handed Weyl spinor?

I know those are very basic questions. Thanks for your help.

blechman
Thank you very much for replying. I feel like this is very helpful but I am still confused a bit.
First, to make sure we are using the same language, I am thinking in terms of Weyl spinors, not Majorana. So the left-handed spinor has two degrees of freedom, matched by the two degrees of freedom of the compelx scalar field.

Hmmm... ok, you have to be careful. I'm not talking about fields, at all - I'm talking about STATES. In particular: if we have a vacuum $|\Omega\rangle$ we can act on it with the susy generator (raising operator):

$$\bar{Q}_{\dot{1}}|\Omega\rangle$$

for MASSLESS states, this is the ONLY thing you can construct - this follows from the algebra (all other Q's kill the vacuum). Then if the vacuum has spin j, this new state has helicity j+1/2. Then you must include the CPT-conjugate states (see below) to complete the multiplet. These states are then described in field theory by a weyl fermion + complex scalar for j=-1/2, and a vector and a weyl fermion for j=-1.

For MASSIVE states, you can also act on the vacuum with $\bar{Q}_{\dot{2}}$ (which you can show from the algebra LOWERS j), so you have four states:

$$|\Omega\rangle$$
$$\bar{Q}_{\dot{1}}|\Omega\rangle$$
$$\bar{Q}_{\dot{2}}|\Omega\rangle$$
$$\bar{Q}_{\dot{1}}\bar{Q}_{\dot{2}}|\Omega\rangle$$

for j=0: the first and last state for the two DOF of the complex scalar, while the middle states have +1/2 (-1/2) for the Weyl fermion. For j=1/2... well, massive vectors violate gauge invariance, so you know the only way to do that is to have a vector, TWO Weyl fermions (with a Majorana mass) and a complex scalar (SUSY Higgs mechanism) - the above gives you the first half of it, while the CPT conjugate (j=-1/2) gives you the other half.

I'm silly - I should have said CPT-conjugate. That's what I meant. Sorry. It's not a question of adding antiparticles: it's a question of not violating the CPT theorem of QFT!

A second related question: the two degrees of freedom of a left-handed Weyl spinor have j=-1/2 and j=+1/2 ? Even though they are both part of a left-handed Weyl spinor?

I know those are very basic questions. Thanks for your help.

Weyl spinors have two degrees of freedom. One can think of them as the LEFT-handed particle, and the RIGHT-handed ANTI-particle, if you like - this is just a consequence of CPT invariance that both must be present in a local, unitary QFT.

This is a great question - it's making me go back and relearn all this stuff! Feel free to ask more. I'm probably butchering the explanation, but any serious text on susy goes through this stuff pretty well. You can check out Wess&Bagger, or John Terning's excellent new (well, relatively new) text. There's also some great web-resources out there. Bookmarked on my web browser is Philip Argyres lectures at http://www.physics.uc.edu/~argyres/661/index.html
They're pretty advanced (Argyres is a big Seiberg-duality type of guy!) but I like them...

Hope this helps!

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I am sorry I am being so slow but I feel like you have the knowledge to clear up everything for me. Just imagine that you are talking to a 5 years old and I will eventually get it!

Let's focus on the massless chiral state to start with. If I can pass the test on that one, I will graduate to the massive and vector multiplet class.

Hmmm... ok, you have to be careful. I'm not talking about fields, at all - I'm talking about STATES. In particular: if we have a vacuum $|\Omega\rangle$ we can act on it with the susy generator (raising operator):

$$\bar{Q}_{\dot{1}}|\Omega\rangle$$

for MASSLESS states, this is the ONLY thing you can construct - this follows from the algebra (all other Q's kill the vacuum). Then if the vacuum has spin j, this new state has helicity j+1/2.

So far so good. But to be clear about counting degrees of freedom: so far (i.e. before including the CPT conjugate states), how many degrees of freedom do we have? Only one bosonic and one fermionic, right? Just making sure.

Then you must include the CPT-conjugate states (see below) to complete the multiplet. These states are then described in field theory by a weyl fermion + complex scalar for j=-1/2, and a vector and a weyl fermion for j=-1.

Weyl spinors have two degrees of freedom. One can think of them as the LEFT-handed particle, and the RIGHT-handed ANTI-particle, if you like - this is just a consequence of CPT invariance that both must be present in a local, unitary QFT.

Ok, this is what I was missing. You are saying that on-shell, a left-handed weyl spinor has only one degree of freedom? And the second degree of freedom added by including the CPT conjugate states is actually the right-handed antiparticle state?

This is where I am a bit confused. I thought, for some reason, that a massless, on-shell Weyl spinor had two degrees of freedom (and I mean not including the antiparticle degree of freedom). I guess I thought that because it seemed to me that a Weyl spinor has two complex components (or, if we were to use Majorana representation there would be 4 real components) and that the equation of motion was cutting that in half. Therefore I thought there were two degrees of freedom for the particle alone.

But I guess that what I was missing is the fact that taking the components to be complex means that I was already including the antiparticle degrees of freedom, correct?

So when we include the CPT conjugate states, we are adding new states which have the opposite helicities to the states we started with? Is that correct?

In the case of a scalar, the initial state plus its CPT conjugate produces a complex scalar field (two dof).

In the case of a Weyl spinor, the initial state plus its CPT conjugate yields the left-handed particle plus the right-handed antiparticle for a total of two dof, right?

In the case of a spin one particle which is its own antiparticle (example the photon), the initial state plus its CPT conjugate yields the two polarization of an on-shell photon, right?

If this is all correct, then everything would make sense. My problem was that I thought that the particle state of the Weyl left-handed spinor had two dof and that messed up completely my understanding.

I will think about the massive states but before getting there, how does it work for a Majorana spinor? It's its own antiparticle, right? But then I am not sure what the CPT conjugate state that we are adding corresponds to...and what role helicity plays here.

And what if we work with N=1 SUSY in D dimensions. I seems as if we would not generate enough states for a photon by simply adding the state to its CPT conjugate state (don't spin one massless on-shell particles have D-2 dof in D spacetime dimensions? I may be wrong about this)

This is a great question - it's making me go back and relearn all this stuff! Feel free to ask more. I'm probably butchering the explanation,
You are helping enormously!!
but any serious text on susy goes through this stuff pretty well. You can check out Wess&Bagger,

I have looked at almost all books on SUSY and I got stuck early on because it felt to me like there was always so much stuff simply thrown at me without giving some motivation. when I learn something new I need to know if something is an educated guess, something tha follows from some principle of symmetry (and if so, what is the pricniple and how does it follow), something that follows from another expression through algebra, etc. when there is too much stuff thrown at me that feel like non sequitur, I can't understand.
or John Terning's excellent new (well, relatively new) text.
I will look up that one, it's new to me.
I finally started to understand SUSY when I found the book (based on a review paper) written by Aitchison. It would probably feel too basic for you but for me it's perfect, I needed a reference that would take my hand and guide me slowly and with justification for everything!
There's also some great web-resources out there. Bookmarked on my web browser is Philip Argyres lectures at http://www.physics.uc.edu/~argyres/661/index.html
They're pretty advanced (Argyres is a big Seiberg-duality type of guy!) but I like them...
He is a very nice guy, he was a postdoc at my university when I was doing my PhD there. I have them but again, when I looked at them a few years ago I got lost. After Aitchison I would probably be able to make sense of them. Thanks for reminding me
Hope this helps!
It helps A LOT. No book can replace exchanges with a live person who can teach you a topic! Thank you so much!

Patrick

...

This is where I am a bit confused. I thought, for some reason, that a massless, on-shell Weyl spinor had two degrees of freedom (and I mean not including the antiparticle degree of freedom). I guess I thought that because it seemed to me that a Weyl spinor has two complex components (or, if we were to use Majorana representation there would be 4 real components) and that the equation of motion was cutting that in half. Therefore I thought there were two degrees of freedom for the particle alone.

But I guess that what I was missing is the fact that taking the components to be complex means that I was already including the antiparticle degrees of freedom, correct?

So when we include the CPT conjugate states, we are adding new states which have the opposite helicities to the states we started with? Is that correct?

In the case of a scalar, the initial state plus its CPT conjugate produces a complex scalar field (two dof).

In the case of a Weyl spinor, the initial state plus its CPT conjugate yields the left-handed particle plus the right-handed antiparticle for a total of two dof, right?

In the case of a spin one particle which is its own antiparticle (example the photon), the initial state plus its CPT conjugate yields the two polarization of an on-shell photon, right?

If this is all correct, then everything would make sense. My problem was that I thought that the particle state of the Weyl left-handed spinor had two dof and that messed up completely my understanding.

...

So going on-shell does indeed cut down two degrees of freedom from the spinor, leaving two that match with the complex scalar. Off-shell, one has four spinor degrees of freedom, so we add a complex scalar auxiliary field $$F$$ with trivial equations of motion $$F = F* = 0$$ to keep the number of fermionic and bosonic degrees of freedom the same.

I'll add Martin's Supersymmetry Primer (hep-ph/9709356, current version is from 2006) to the list of recommendations, so that I can quote from his discussion in Section 3.1:
In retrospect, one can see why we needed to introduce the auxiliary field F in order to get the supersymmetry algebra to work off-shell. On-shell, the complex scalar field φ has two real propagating degrees of freedom, matching the two spin polarization states of $$\psi$$. Off-shell, however, the Weyl fermion $$\psi$$ is a complex two-component object, so it has four real degrees of freedom. (Going on-shell eliminates half of the propagating degrees of freedom for $$\psi$$, because the Lagrangian is linear in time derivatives, so that the canonical momenta can be reexpressed in terms of the configuration variables without time derivatives and are not independent phase space coordinates.) To make the numbers of bosonic and fermionic degrees of freedom match off-shell as well as on-shell, we had to introduce two more real scalar degrees of freedom in the complex field $$F$$, which are eliminated when one goes on-shell. This counting is summarized in Table 3.1. The auxiliary field formulation is especially useful when discussing spontaneous supersymmetry breaking, as we will see in section 6.
The spin-one case is similar. As you say, on-shell (massless), there are two degrees of freedom for both the vector and spinor. Going off-shell (massive), there are three spin-one degrees of freedom and four spin-one-half degrees of freedom, so we add a single real boson auxiliary field $$D$$ (Martin, Section 3.3).

I'm actually surprised Martin's primer hadn't been mentioned yet. Peskin's 2006 TASI lecture notes, "Supersymmetry in Elementary Particle Physics" (0801.1928) may also be useful.

blechman
So far so good. But to be clear about counting degrees of freedom: so far (i.e. before including the CPT conjugate states), how many degrees of freedom do we have? Only one bosonic and one fermionic, right? Just making sure.

Hmmm... yes, this is correct, but...

Ok, this is what I was missing. You are saying that on-shell, a left-handed weyl spinor has only one degree of freedom? And the second degree of freedom added by including the CPT conjugate states is actually the right-handed antiparticle state?

this is where your problem is: give up on the notion of a "Weyl spinor" for the moment - that's WAY beyond a 5-year-old's vocabulary!

I'm doing nothing but ordinary quantum MECHANICS at this stage in the game. We have identified one spin-1/2 DOF and one spin-0 DOF, as you mention above. Now, one spin-0 DOF is described just fine by a real scalar field, but ONE spin-1/2 DOF is *NOT* described by anything you know of, since there is no such thing as a single, isolated spin-1/2 DOF in nature (because of the CPT theorem). Spin-1/2 (in 3+1 dimensions) needs a MINIMUM of TWO DOF's to satisfy CPT theorem (you can think of this as the LH particle and RH antiparticle, which *MUST* appear together in any local, Lorentz-invariant, unitary QFT). THIS is the Weyl spinor that you are looking for. But in order to have two DOF that are related by CPT, we must include the CPT conjugate STATE. Then the Weyl spinor describes the two states, and the two leftover spin-0 DOF marry quite nicely into a COMPLEX scalar field. Now, and ONLY now(!), life is good.

This is where I am a bit confused. I thought, for some reason, that a massless, on-shell Weyl spinor had two degrees of freedom (and I mean not including the antiparticle degree of freedom). I guess I thought that because it seemed to me that a Weyl spinor has two complex components (or, if we were to use Majorana representation there would be 4 real components) and that the equation of motion was cutting that in half. Therefore I thought there were two degrees of freedom for the particle alone.

But I guess that what I was missing is the fact that taking the components to be complex means that I was already including the antiparticle degrees of freedom, correct?

Exactly! Everything you said above (except for the "not including antiparticle...") is correct.

So when we include the CPT conjugate states, we are adding new states which have the opposite helicities to the states we started with? Is that correct?

Quite. That's the "P" part of "CPT".

In the case of a scalar, the initial state plus its CPT conjugate produces a complex scalar field (two dof).

Well, in general you don't need two spin-0 modes to make a CPT-invariant theory, but you need two fermionic modes, and then SUSY means that you need two scalar modes (a complex scalar). That would be the more correct way of saying it. But what you're saying is basically correct.

In the case of a Weyl spinor, the initial state plus its CPT conjugate yields the left-handed particle plus the right-handed antiparticle for a total of two dof, right?

Which, TOGETHER, make the Weyl spinor. Independently, they make trouble!

In the case of a spin one particle which is its own antiparticle (example the photon), the initial state plus its CPT conjugate yields the two polarization of an on-shell photon, right?

Correct.

If this is all correct, then everything would make sense. My problem was that I thought that the particle state of the Weyl left-handed spinor had two dof and that messed up completely my understanding.

No, the problem is that you're actually not a 5-year-old! You're thinking in terms of QFT with Weyl spinors, etc, and this is the younger QM, which makes no use of the notion of Weyl fields. The key is that the state $Q_{\dot{1}}|\Omega_0\rangle$ is NOT the Weyl spinor, it's just one of the two states that the Weyl spinor will annihilate in the Fock-space language of QFT.

I will think about the massive states but before getting there, how does it work for a Majorana spinor? It's its own antiparticle, right? But then I am not sure what the CPT conjugate state that we are adding corresponds to...and what role helicity plays here.

One thing at a time. This is for another post, but I'll get you started: A WEYL spinor ($\xi$) is a two component spinor. A DIRAC component spinor is *TWO* Weyl spinors placed into a 4-component field $(\xi,\bar{\chi})$, where the bar is basically complex conjugation. A MAJORANA spinor is a 4-component Dirac spinor with $\xi=\chi$, so in fact it is SECRETLY only a Weyl spinor written in a 4D representation. A fermion mass in terms of Weyl spinors is a mass term of the form:

$$m(\bar{\xi}\chi + \bar{\chi}\xi)[/itex] if $\chi=\xi$ it's called a MAJORANA mass; if the two spinors are not equal, it's called a DIRAC mass. From this, you can see many statements of particle physics win out: Dirac mass terms (coupling two spinors that transform differently under the EW gauge group) are not allowed without a Higgs; a Majorana mass is forbidden by gauge invarance of all fields except a neutrino (which is electrically neutral) - again with the help of the Higgs. Also, if RH neutrinos do not exist (no $\bar{\chi}$ for a neutrino field) then you CANNOT have Dirac masses for neutrinos; etc. We can go on and on about spinor field theory, but I won't. As to massive multiplets: again, don't think in terms of QFT, but in terms of QM. Notice that there are now more states we can write down (see my previous post for the list) and just follow the countings. Don't implement the QFT until you understand what the Fock space looks like. And what if we work with N=1 SUSY in D dimensions. I seems as if we would not generate enough states for a photon by simply adding the state to its CPT conjugate state (don't spin one massless on-shell particles have D-2 dof in D spacetime dimensions? I may be wrong about this) OH, BOY! Now you've taken the plunge! Since, as you point out, the spinor reps depend very sensitively on the dimension, it is certainly clear that the game has changed considerably! For example, the 5D Poincare group does NOT allow for Weyl representations (without a gamma_5 the Dirac rep is no longer reducible) so the Q's are now DIRAC (4-component) spinors. This turns out to be equivalent to TWO Weyl spinors with (potentially) nontrival commutation between them. This is called "D=4,N=2 SUSY" - and it gets worse from here. String (M) theory calling for D=10 (11) can be represented by the supergravity multiplet of D=4,N=8, which just so happens to be the maximum you can go; what a funny "coincidence" that the critical dimension of superstring/M theory is the same as the maximum dimension that has an effective 4D description! Let's not go down this route until we've worked out D=4,N=1. I think that would be wise! I have looked at almost all books on SUSY and I got stuck early on because it felt to me like there was always so much stuff simply thrown at me without giving some motivation. when I learn something new I need to know if something is an educated guess, something tha follows from some principle of symmetry (and if so, what is the pricniple and how does it follow), something that follows from another expression through algebra, etc. when there is too much stuff thrown at me that feel like non sequitur, I can't understand. Yeah, you have to be motivated first, and THEN start studying it. Well, SUSY solves the hierarchy problem, may help unification, if you focus on local, unitary QFT with an S-matrix it's the only way to generalize the Poincare algebra (Coleman-Mandula). If this doesn't do it for you, then I don't know what will. Heck, I'm not sure it does it for me either, and I make a career out of this crap! :tongue2: I will look up that one, it's new to me. I finally started to understand SUSY when I found the book (based on a review paper) written by Aitchison. It would probably feel too basic for you but for me it's perfect, I needed a reference that would take my hand and guide me slowly and with justification for everything! I don't know that one, but I'm happy you got something out of it. If you don't want to shell out the money for John's book (it's a good investment for those serious about SUSY), you can check out his TASI-2002 lectures, on which the book is based. But they might not have what you want (they're about the nonperturbative SUSY stuff - quite fascinating, but not what you asked about, and WAY beyond a 5-year-old!!). It helps A LOT. No book can replace exchanges with a live person who can teach you a topic! Thank you so much! Patrick I'm happy I was able to help. Keep asking... So going on-shell does indeed cut down two degrees of freedom from the spinor, leaving two that match with the complex scalar. Off-shell, one has four spinor degrees of freedom, so we add a complex scalar auxiliary field [tex]F$$ with trivial equations of motion $$F = F* = 0$$ to keep the number of fermionic and bosonic degrees of freedom the same.

I'll add Martin's Supersymmetry Primer (hep-ph/9709356, current version is from 2006) to the list of recommendations, so that I can quote from his discussion in Section 3.1:

The spin-one case is similar. As you say, on-shell (massless), there are two degrees of freedom for both the vector and spinor. Going off-shell (massive), there are three spin-one degrees of freedom and four spin-one-half degrees of freedom, so we add a single real boson auxiliary field $$D$$ (Martin, Section 3.3).

This is all quite true, but irrelevant at this stage. Everything above is ON-shell, so there are no auxiliary fields. The problem you mention comes when you go OFF-shell, and the two complex DOF of the Weyl spinor become FOUR real DOF, which is too many. Then you need to compensate with auxiliary fields. However, on-shell you still have TWO degrees of freedom, and the SINGLE state $Q_{\dot{1}}|\Omega_0\rangle$ is only one of them. That's not the same as the introduction of auxiliary fields.

I'm actually surprised Martin's primer hadn't been mentioned yet. Peskin's 2006 TASI lecture notes, "Supersymmetry in Elementary Particle Physics" (0801.1928) may also be useful.

Yes, people I've worked with have a real love-hate relationship with Martin's primer, which is why I didn't mention it. I didn't particularly care much for his early sections on doing SUSY, and I don't remember if he did this SUSY-QM stuff. What I *LOVE* Martin's notes for is the latter sections where he does the MSSM itself. Even now, after doing this stuff for years, I still use his sections 7-10 as a desk reference for when I need to look something up.

As for Peskin's TASI lectures - I didn't read them, but he's a pretty good lecturer so I'm sure they're a good place to look as well.

Hope I didn't muddle things up -- that's just what the d.o.f. discussion brought up in my mind. I just did a reading course on SUSY using Peskin, Martin and Argyres this Spring, but didn't learn it as well as I should have (so I'm finding this discussion very interesting). I stumbled across Aitchison's review (hep-ph/0505105) towards the end of the semester and printed it out (in a somewhat condensed format), but haven't been able to go through it yet.

I think "motivation" is being used here more in the sense of "logical development" than "reasons to study this stuff in the first place", and Aitchison does look to devote a lot of attention to that. This is why I'm still hoping to go through his review even if it is "more basic" than the others.

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Hope I didn't muddle things up -- that's just what the d.o.f. discussion brought up in my mind. I just did a reading course on SUSY using Peskin, Martin and Argyres this Spring, but didn't learn it as well as I should have (so I'm finding this discussion very interesting). I stumbled across Aitchison's review (hep-ph/0505105) towards the end of the semester and printed it out (in a somewhat condensed format), but haven't been able to go through it yet.

I think "motivation" is being used here more in the sense of "logical development" than "reasons to study this stuff in the first place", and Aitchison does look to devote a lot of attention to that. This is why I'm still hoping to go through his review even if it is "more basic" than the others.

Yes, you understood what I meant by motivation. You might find Aitchison too elementary but fo rme it was *perfect*. He motivates everything very well. It always bugged me to be given the algebra of the charges as a starting point like most books/reviews do because they were not motivated. When something seems to be handed down by God, I don't feel like I understand it. I have to see the logic behind something before I feel that I really understand something. Aitchison presents the SUSY transformations in a way that probably most people in the field would find way too tedious and roundabout but it's exactly what I needed because every single step has some logic too it! THAT is the way I would teach the subject to students because I could, as a prof, justify everything.

After going through Aitchison I looked at Martin and then it all made sense!

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Thanks again A LOT for your help.

Hmmm... yes, this is correct, but...

this is where your problem is: give up on the notion of a "Weyl spinor" for the moment - that's WAY beyond a 5-year-old's vocabulary!

I'm doing nothing but ordinary quantum MECHANICS at this stage in the game.
Ok. "Fields" came to mind only because I associated "antiparticle states" to quantum field theory! That's the only reason I talked about fields. So the question is: can we talk about requiring a theory to have CPT invariance without at the same time imposing Lorentz invariance and therefore introducing quantum fields? Or is it consistent to require CPT invariance on a non-relativisitic QM theory? But I don't want this to be a distraction from SUSY so feel free toignore this question!
We have identified one spin-1/2 DOF and one spin-0 DOF, as you mention above. Now, one spin-0 DOF is described just fine by a real scalar field, but ONE spin-1/2 DOF is *NOT* described by anything you know of, since there is no such thing as a single, isolated spin-1/2 DOF in nature (because of the CPT theorem).
Yes, that makes perfect sense. And I knew that...except that I did not think that this second dof had anything to do with an antiparticle state
Spin-1/2 (in 3+1 dimensions) needs a MINIMUM of TWO DOF's to satisfy CPT theorem (you can think of this as the LH particle and RH antiparticle, which *MUST* appear together in any local, Lorentz-invariant, unitary QFT).
It makes sense but the last bit of confusion left in my head is related to the part you can think of this as.. I don't know if this is meant to be a very rough analogy (or a cartoon) or if it is meant to be litteral. So let me be sure: this other degree of freedom IS actually the antiparticle right-handed state, isn't? I mean by that that when people say that a Weyl spinor has two dof, the antiparticle state is included in that counting, correct?
I know that I am annoyingly repeating questions but I just want to be sure I get it right.
THIS is the Weyl spinor that you are looking for. But in order to have two DOF that are related by CPT, we must include the CPT conjugate STATE. Then the Weyl spinor describes the two states, and the two leftover spin-0 DOF marry quite nicely into a COMPLEX scalar field. Now, and ONLY now(!), life is good.
yes, life is good now that you helped me quite a bit to understand better.
.....

Exactly! Everything you said above (except for the "not including antiparticle...") is correct.

.....

Quite. That's the "P" part of "CPT".

.....

Well, in general you don't need two spin-0 modes to make a CPT-invariant theory, but you need two fermionic modes, and then SUSY means that you need two scalar modes (a complex scalar). That would be the more correct way of saying it. But what you're saying is basically correct.

.....

Which, TOGETHER, make the Weyl spinor. Independently, they make trouble!

......

Correct.

......

No, the problem is that you're actually not a 5-year-old! You're thinking in terms of QFT with Weyl spinors, etc, and this is the younger QM, which makes no use of the notion of Weyl fields. The key is that the state $Q_{\dot{1}}|\Omega_0\rangle$ is NOT the Weyl spinor, it's just one of the two states that the Weyl spinor will annihilate in the Fock-space language of QFT.

It's probably my fault for using incorrect terminology. The only reason I used "fields" was that I was seeing you talk about including antiparticle states. And when I see particles together with their antiparticles I think "fields" automatically. But I see your point.

One thing at a time. This is for another post, but I'll get you started: A WEYL spinor ($\xi$) is a two component spinor. A DIRAC component spinor is *TWO* Weyl spinors placed into a 4-component field $(\xi,\bar{\chi})$, where the bar is basically complex conjugation. A MAJORANA spinor is a 4-component Dirac spinor with $\xi=\chi$, so in fact it is SECRETLY only a Weyl spinor written in a 4D representation. A fermion mass in terms of Weyl spinors is a mass term of the form:

$$m(\bar{\xi}\chi + \bar{\chi}\xi)[/itex] if $\chi=\xi$ it's called a MAJORANA mass; if the two spinors are not equal, it's called a DIRAC mass. Thanks for the refresher. I am relatively ok with this. My problem was with the SUSY states generated by the SUSY charge when one works with a Majorana fermion instead of a Weyl fermion (in D=4, for a massless chiral multiplet). So we have a state with helicity = -1/2 which corresponds to one of of the dof of the Majorana spinor. Now the CPT conjugate will involve a state with helicity of +1/2. Now, in the Weyl language, you said that this second state could be thought as the antiparticle right-handed state. But a Majorana spinor is its own antiparticle. So what is the proper point of view here? A Majorana spinor has two degrees of freedom which have opposite helicities? From this, you can see many statements of particle physics win out: Dirac mass terms (coupling two spinors that transform differently under the EW gauge group) are not allowed without a Higgs; a Majorana mass is forbidden by gauge invarance of all fields except a neutrino (which is electrically neutral) - again with the help of the Higgs. Also, if RH neutrinos do not exist (no $\bar{\chi}$ for a neutrino field) then you CANNOT have Dirac masses for neutrinos; etc. We can go on and on about spinor field theory, but I won't. As to massive multiplets: again, don't think in terms of QFT, but in terms of QM. Notice that there are now more states we can write down (see my previous post for the list) and just follow the countings. Don't implement the QFT until you understand what the Fock space looks like. Thanks again, all very interesting. I will get back to the massive states soon OH, BOY! Now you've taken the plunge! Since, as you point out, the spinor reps depend very sensitively on the dimension, it is certainly clear that the game has changed considerably! For example, the 5D Poincare group does NOT allow for Weyl representations (without a gamma_5 the Dirac rep is no longer reducible) so the Q's are now DIRAC (4-component) spinors. This turns out to be equivalent to TWO Weyl spinors with (potentially) nontrival commutation between them. So even a 5D massless Dirac spinor can't be split into two Weyl spinors? You know the argument that for a massless spinor with its spin in the direction of motion, no observer can overtake it and therefore the helicity is a good quantum number, so that states with opposite helicities do not mix. Is this argument invalid in 5D? Or am I completely mixing up unrelated stuff? This is called "D=4,N=2 SUSY" - and it gets worse from here. I knew that Weyl spinors did not appear in 5D but I thought that Majorana spinors did (but I may easily be wrong about that, I have to check!) This is why I thought that maybe N=1 would still be possible in 5D. Let me make sure I follow. Ok, let's start with the fact that the spinors must be Dirac. Why does it imply that N must be 2? How does it work in terms of dof and helicities? It seems to me that the following will happen: We will have a state with helicity of -1/2. Now, since N=2, we can go up to helicity of +1/2. Adding the CPT conjugate, we have the helciity of +1/2 and -1/2 of the conjugates for a total of 4 dof (that works out in 4D and I do think it's still 4 dof in 5D but I am not sure). Is that correct? SO N can't be equal to one in 5D? String (M) theory calling for D=10 (11) can be represented by the supergravity multiplet of D=4,N=8, which just so happens to be the maximum you can go; what a funny "coincidence" that the critical dimension of superstring/M theory is the same as the maximum dimension that has an effective 4D description! I know! It's in order to understand this that I decided to learn SUSY (as well as understanding BPS states, Seiberg-Witten duality, the AdS/CFT conjecture, and all that exciting stuff. SUSY was all over the place so I knew I had to learn it) Let's not go down this route until we've worked out D=4,N=1. I think that would be wise! I agree! Yeah, you have to be motivated first, and THEN start studying it. Well, SUSY solves the hierarchy problem, may help unification, if you focus on local, unitary QFT with an S-matrix it's the only way to generalize the Poincare algebra (Coleman-Mandula). If this doesn't do it for you, then I don't know what will. Heck, I'm not sure it does it for me either, and I make a career out of this crap! :tongue2: But I think you misunderstood my use of the word "motivation". I am highly motivated in learning that stuff as I explained above. What I meant is that I found that all the books I was reading were not motivating the equations and steps they were making, at least not enough for my taste! It felt like too much stuff was thworn at me without explaining where it came from. That was frustrating me greatly. You would surely find teh book by AItchison too elementary but it's perfect for a five year old child!!! he derivation of the SUSY transformations and the SUSY algebra would probably look WAY too long and tortuous to you, but for me it was perfect because every step of teh way made sense to me. It was long and roundabout but every step made perfect sense. I prefer that to be given an algebra that seems to come from nowhere and a bunch of notation (dotted, undotted, etc) without motivation behind it. SO Aitchison saved me from giving up on learning SUSY. I don't know that one, but I'm happy you got something out of it. If you don't want to shell out the money for John's book (it's a good investment for those serious about SUSY), you can check out his TASI-2002 lectures, on which the book is based. But they might not have what you want (they're about the nonperturbative SUSY stuff - quite fascinating, but not what you asked about, and WAY beyond a 5-year-old!!). I checked the book you mentioned but it's unfortunately 107. I will look at the TASI lectures. Thanks After reading Aitchison (part of it, I am not done yet), Martin's review paper made much more sense. I'm happy I was able to help. Keep asking... I sure appreciate greatly your help!!! Thanks for the offer, I will have more questions. I am trying to learn about all the exciting stuff....string, SUGRA, SUSY, dualities, black hole entropy stuff, loop quantum gravity, brane worlds, and on and on. In six months I will take a two year unpaid sabbatical and try to find someone to do research with on any of these topics, free of charge (i.e. I will pay all my expenses, I will only ask a desk to sit and someone to work with on a project) Although I'm not interefering, I'd like to thank you guys, because this is indeed very interesting, I'm sure for many of us. blechman Science Advisor Ok. "Fields" came to mind only because I associated "antiparticle states" to quantum field theory! That's the only reason I talked about fields. So the question is: can we talk about requiring a theory to have CPT invariance without at the same time imposing Lorentz invariance and therefore introducing quantum fields? Or is it consistent to require CPT invariance on a non-relativisitic QM theory? But I don't want this to be a distraction from SUSY so feel free toignore this question! I don't remember what the precise statement of the CPT theorem is - something about CPT invariance is necessary for local, unitary theory. I don't remember if Lorentz-invariance is part of the mix or not. But let's leave this question for the time being and push on. If anyone out there has good explanation, please feel free to post it. Yes, that makes perfect sense. And I knew that...except that I did not think that this second dof had anything to do with an antiparticle state ... It makes sense but the last bit of confusion left in my head is related to the part you can think of this as.. I don't know if this is meant to be a very rough analogy (or a cartoon) or if it is meant to be litteral. So let me be sure: this other degree of freedom IS actually the antiparticle right-handed state, isn't? I mean by that that when people say that a Weyl spinor has two dof, the antiparticle state is included in that counting, correct? I know that I am annoyingly repeating questions but I just want to be sure I get it right. No, no cartooning! This is PRECISELY what the two degrees of freedom of the Weyl spinor are. It's probably my fault for using incorrect terminology. The only reason I used "fields" was that I was seeing you talk about including antiparticle states. And when I see particles together with their antiparticles I think "fields" automatically. But I see your point. I know I have been rather ridiculous about the fields vs quantum mechanics, but in this case it's an important distinction. Since everything we're doing is special-relativistic QM, so we should really be using the language of QFT: I am talking about the "Fock space" that the Weyl field will ultimately act on. I wanted to make sure you understood that this is NOT the same as the Weyl field itself, although in practice we rarely-to-never make the distinction. I know you know this stuff, but especially for those that might be listening in, I hope I made that clear. Thanks for the refresher. I am relatively ok with this. My problem was with the SUSY states generated by the SUSY charge when one works with a Majorana fermion instead of a Weyl fermion (in D=4, for a massless chiral multiplet). So we have a state with helicity = -1/2 which corresponds to one of of the dof of the Majorana spinor. Now the CPT conjugate will involve a state with helicity of +1/2. Now, in the Weyl language, you said that this second state could be thought as the antiparticle right-handed state. But a Majorana spinor is its own antiparticle. So what is the proper point of view here? A Majorana spinor has two degrees of freedom which have opposite helicities? There is no difference between a "Majorana fermion" and a "Weyl fermion" (in D=4) - that's what I was trying to say before. You have to be careful about statements like: "Majorana is its own antiparticle" - that statement is misleading. What Majorana REALLY means is that there is a Majorana mass term. Let's take neutrinos (the canonical example from particle physics): there are two kinds of mass terms one can write down: [tex]\nu_L\nu_L + \bar{\nu}_L\bar{\nu}_L$$

and

$$\bar{\nu}_L\nu_R + \bar{\nu}_R\nu_L$$

(assuming there IS a $\nu_R$. Both of these terms involve two-component Weyl spinors. The first is a Majorana mass, the second is a Dirac mass. Recall that things like

$$\bar{\nu}_L\nu_L$$

are not Lorentz invariant, so that's all we can write down. Now the Majorana mass, as you can see, violates lepton number by 2 units, and so has the EFFECT of "mixing" the (left-handed) neutrino and (right-handed) antineutrino. Thus we make the cavalier statement that "a Majorana neutrino is its own antiparticle" - what me mean is that the neutrino and antineutrino "mix", and thus allow for lepton number violation.

So bringing it home - don't think of a Majorana fermion as "its own antiparticle;" think of it as it is: a mass term that violates a particle number symmetry.

Thanks again, all very interesting. I will get back to the massive states soon

So even a 5D massless Dirac spinor can't be split into two Weyl spinors? You know the argument that for a massless spinor with its spin in the direction of motion, no observer can overtake it and therefore the helicity is a good quantum number, so that states with opposite helicities do not mix. Is this argument invalid in 5D? Or am I completely mixing up unrelated stuff?

the short, albeit unenlightened answer is that the little group of massless 5D is no longer SO(2), so helicity does not exist! I will leave it at this for now (since it's getting off-topic) but we can come back and discuss higher-dimensional Lorentz group reps later.

I knew that Weyl spinors did not appear in 5D but I thought that Majorana spinors did (but I may easily be wrong about that, I have to check!) This is why I thought that maybe N=1 would still be possible in 5D.

Nope - there are no Majorana spinors in odd dimensions, only Dirac. Therefore a SINGLE spinor in D=5 (like the susy generators) must be a Dirac spinor, which, as I said before, is equivalent to two Weyl spinors. Therefore we have that:

(D=5,N=1) <==> (D=4,N=2)

Notice that we still say N=1 in the 5D description since there is ONE susy generator, but in terms of 4D fields (relevant if you were to compactify the extra dimension, for example) it reduces to N=2.

Let me make sure I follow. Ok, let's start with the fact that the spinors must be Dirac. Why does it imply that N must be 2? How does it work in terms of dof and helicities?

It seems to me that the following will happen: We will have a state with helicity of -1/2. Now, since N=2, we can go up to helicity of +1/2. Adding the CPT conjugate, we have the helciity of +1/2 and -1/2 of the conjugates for a total of 4 dof (that works out in 4D and I do think it's still 4 dof in 5D but I am not sure).

Is that correct? SO N can't be equal to one in 5D?

Again, when working in D=5, we still have N=1, but the states no longer have an SO(2) helicity since that's not the little group anymore. The game changes. I'm gonna leave it at that for now, but if you want to push the issue you can ask me. Let's try to nail down D=4 first, though.

I know! It's in order to understand this that I decided to learn SUSY (as well as understanding BPS states, Seiberg-Witten duality, the AdS/CFT conjecture, and all that exciting stuff. SUSY was all over the place so I knew I had to learn it)

Some famous person (Shamit Kachru, I think, but I"m not sure) once said to me: SUSY is the theorist's laboratory! It's where we have enough power to actually compute things. Even if it has no relevance in the "real world" that's enough to make it worth studying, like $\phi^4$ theory in QFT. I think your comment above exactly makes this point!

But I think you misunderstood my use of the word "motivation". I am highly motivated in learning that stuff as I explained above.
What I meant is that I found that all the books I was reading were not motivating the equations and steps they were making, at least not enough for my taste! It felt like too much stuff was thworn at me without explaining where it came from. That was frustrating me greatly. You would surely find teh book by AItchison too elementary but it's perfect for a five year old child!!! he derivation of the SUSY transformations and the SUSY algebra would probably look WAY too long and tortuous to you, but for me it was perfect because every step of teh way made sense to me. It was long and roundabout but every step made perfect sense. I prefer that to be given an algebra that seems to come from nowhere and a bunch of notation (dotted, undotted, etc) without motivation behind it. SO Aitchison saved me from giving up on learning SUSY.

I checked the book you mentioned but it's unfortunately 107\$.
I will look at the TASI lectures. Thanks

After reading Aitchison (part of it, I am not done yet), Martin's review paper made much more sense.

Yeah, John's book is offensively expensive! I already yelled at him about it, as if he can do anything . I got it when it was on sale. But the TASI lectures are also good.

I sure appreciate greatly your help!!! Thanks for the offer, I will have more questions.

I am trying to learn about all the exciting stuff....string, SUGRA, SUSY, dualities, black hole entropy stuff, loop quantum gravity, brane worlds, and on and on. In six months I will take a two year unpaid sabbatical and try to find someone to do research with on any of these topics, free of charge (i.e. I will pay all my expenses, I will only ask a desk to sit and someone to work with on a project)

You know who you should talk to is your old pal, Cliff Burgess. I seriously don't know many people who knows as much about as much as he does! And to top it off, he's really friendly and willing to explain things! And he has a nack for coming up with amazing ideas (at least I think so). Anyway, if you haven't already, you should drop him a line.

You have quite an impressive list of study-topics! I showed it to my colleague here (who has helped confirm some of the stuff I've been saying, especially about the Majorana fermions), and his (excellent) advice is to pick one or two of those, and stick to that! I'm sure you don't need anyone to tell you how time consuming this stuff can be. But I wish you good luck, and I'd be happy to help you in what I know (not so much as you leave the weak scale behind...).

It's ironic: you are thinking about pushing back into research, while I am considering going into full-time teaching!

Homework Helper
Gold Member
I don't remember what the precise statement of the CPT theorem is - something about CPT invariance is necessary for local, unitary theory. I don't remember if Lorentz-invariance is part of the mix or not. But let's leave this question for the time being and push on. If anyone out there has good explanation, please feel free to post it.

No, no cartooning! This is PRECISELY what the two degrees of freedom of the Weyl spinor are
Thanks again Blechman for your 1 over epsilon patience.

It seemed completely clear...but then a nagging doubt raised its ugly head in my mind and I realized that something was still bugging me. Let me try to explain (now you WILL be convinced that I am five years old!)

The way I think of a left-handed Weyl spinor here is what comes to mind: it is a two component spinor which transforms under an infinitesimal Lorentz transformation according to something schematically of the form (I never remember the correct signs and the factors of i so that's just the basic structure):

$$\psi_L \rightarrow (1 - i \xi \cdot \sigma + \epsilon \cdot \eta ) \psi_L$$
where the two terms correspond to rotations and boosts, respectively.

The two component spinor contains 2 complex parameters which are halved upon imposing the equation of motion. So this leaves two dof.

This state has a specific helicity. If I apply the helicity operator to it I get minus times the state.

I realize that I am still a bit confused. It's because of the above that in my mind, the particle state had by itself two degrees of freedom! Since the two component spinor has two parameters (after imposing the eom) and the whole thing is left-handed.
So it seemed to me that the *left-handed particle state by itself* had two dof.

However you point out that there is only one dof associated to the left-handed particle state and that a second dof actually comes from the right-handed antiparticle state.

I am trying to reconcile this with the above picture of a left-handed spinor simply being a two-component column vector that is an eigenstate of the helicity operator.

Until you brought it up, I had never thought about the antiparticle degrees of freedom, so I used to think that when we said that there are two fermionic dof in a chiral multiplet, those were the two dof of the left-handed two component spinor described above.

Now, if I want to include the CPT conjugate of the above, two component, spinor, I can imagine that what will happen is that the parameters of the CPT conjugate will be determined by the components of the left-handed particle so I could see how we could still end up with two dof even after including the CPT conjugate.

But it is still difficult for me to see that we can assign one dof to the left-handed particle state and one dof to the right-handed antiparticle state.

The only way I could imagine to make the connection with the way you explain it would be if there is a way to somehow do some type of linear transformation on the parameters of my left-handed two component particle Weyl spinor such that only one indepedent parameter ia actually left and that under CPT, this parameter is mapped to a totally independent parameter. I am not sure how to explain it clearly.

But the point is that my left-handed particle spinor seems to me to have two parameters so two dof. And I am trying to reconcile this with your explanation that the helicity=-1/2 state of the chiral multiplet is only one dof and accounts entirely for the particle state.

I know I have been rather ridiculous about the fields vs quantum mechanics, but in this case it's an important distinction. Since everything we're doing is special-relativistic QM, so we should really be using the language of QFT: I am talking about the "Fock space" that the Weyl field will ultimately act on. I wanted to make sure you understood that this is NOT the same as the Weyl field itself, although in practice we rarely-to-never make the distinction. I know you know this stuff, but especially for those that might be listening in, I hope I made that clear.
You were not being ridiculous at all. I think we understood each other all along. I was being quite sloppy with the language, I realize now. I use ''fields'' when talking about degrees of freedom as meaning the different one-particle states that can be created or annihilated by the field so we are in synch here. I was being sloppy.

There is no difference between a "Majorana fermion" and a "Weyl fermion" (in D=4) - that's what I was trying to say before. You have to be careful about statements like: "Majorana is its own antiparticle" - that statement is misleading.

I will wait until I understand the dof of the Weyl spinor to get back to a question about this. I think of a Majorana spinor as being a four-component spinor which obeys a special condition, that the charge conjugation operator applied to it gives the spinor back. I had never thought aout it in terms of what type of mass term was written! I want to get back to this intriguing point later.
the short, albeit unenlightened answer is that the little group of massless 5D is no longer SO(2), so helicity does not exist! I will leave it at this for now (since it's getting off-topic) but we can come back and discuss higher-dimensional Lorentz group reps later.
That would be really be great if we could discuss this topic later.

Nope - there are no Majorana spinors in odd dimensions, only Dirac. Therefore a SINGLE spinor in D=5 (like the susy generators) must be a Dirac spinor, which, as I said before, is equivalent to two Weyl spinors. Therefore we have that:

(D=5,N=1) <==> (D=4,N=2)

Notice that we still say N=1 in the 5D description since there is ONE susy generator, but in terms of 4D fields (relevant if you were to compactify the extra dimension, for example) it reduces to N=2.

Again, when working in D=5, we still have N=1, but the states no longer have an SO(2) helicity since that's not the little group anymore. The game changes. I'm gonna leave it at that for now, but if you want to push the issue you can ask me. Let's try to nail down D=4 first, though.

I am looking forward to it.

Some famous person (Shamit Kachru, I think, but I"m not sure) once said to me: SUSY is the theorist's laboratory! It's where we have enough power to actually compute things. Even if it has no relevance in the "real world" that's enough to make it worth studying, like $\phi^4$ theory in QFT. I think your comment above exactly makes this point!
A very nice way to put it.

No-renormalization theorems are amazing. I read an article where theer was an interesting quote (I am not sure if it's Duff or Schwarz) saying something to the effect that having no renormalization is a theorist dream since it allows to obtain exact results from perturbative calculations!!
Yeah, John's book is offensively expensive! I already yelled at him about it, as if he can do anything . I got it when it was on sale. But the TASI lectures are also good.

You know who you should talk to is your old pal, Cliff Burgess. I seriously don't know many people who knows as much about as much as he does! And to top it off, he's really friendly and willing to explain things! And he has a nack for coming up with amazing ideas (at least I think so). Anyway, if you haven't already, you should drop him a line.
Wow, I am very impressed that you remembered my connection with him!
yes, this is indeed a very good idea. I am a bit intimidated by him and I feel bad because I was going through a very difficult period in my life when I joined his group as a postdoc and I am sure I did not impress him very much. I feel like I need to get a few good papers out to prove myself a bit before contacting him, but that's a catch 22 since it's very hard to publish papers when having been out of the field for a while and having nobody at all to interact with!

Your suggestion is a very good one!

You have quite an impressive list of study-topics! I showed it to my colleague here

He/She must had said ''this guy is nuts!!!

Yes, it's a huge list and each of these topics could take years to master. I am not going in depth, obviously. What I meant is that I am trying to gather some basic background in each topic, just enough that I have some basic understanding of the concepts and of the calculations involved. It just feels sometimes that without a good background in susy, string (including branes), CFT and a few otherthings, it's almost impossible to understand anything at all that is being done. Most papers I read will quickly quote some results from these fields in the first few paragraphs of the paper. So it's frustrating not to be able to understand what is going on.

(who has helped confirm some of the stuff I've been saying, especially about the Majorana fermions), and his (excellent) advice is to pick one or two of those, and stick to that! I'm sure you don't need anyone to tell you how time consuming this stuff can be.

You (and your colleague) are completely right. It's just that it feels like everything is interconnected. Even if I start by focusing on one topic, it quickly feels like so many other things are required. A paper on brane worlds will refer to some result from brane stuff from string theory...to understand that stuff will require to understand BPS states and SUSY...and some other result will be a calculation done using CFT and so on.
The whole field has become monstrous!

You are completely right and I realize that my problem is hat I am trying to get up to speed all my myself so that I don't know what to set aside and what to focus on. I basically would need to do another PhD with an adviser guiding me....

But I wish you good luck, and I'd be happy to help you in what I know (not so much as you leave the weak scale behind...).
Thank you, your help is so much appreciated!
It's ironic: you are thinking about pushing back into research, while I am considering going into full-time teaching!
Really!
At what level? I found teaching fun at first....but after a few years of full time teaching at a low level, I am going crazy I must admit. Teaching at a high level is great fun. And the years I put all research aside made me quite depressed.

Thanks again for all your help!

Patrick

blechman
It seemed completely clear...but then a nagging doubt raised its ugly head in my mind and I realized that something was still bugging me. Let me try to explain (now you WILL be convinced that I am five years old!)

not at all!

I realize that I am still a bit confused. It's because of the above that in my mind, the particle state had by itself two degrees of freedom! Since the two component spinor has two parameters (after imposing the eom) and the whole thing is left-handed.
So it seemed to me that the *left-handed particle state by itself* had two dof.

but ask yourself: WHAT do those two degrees of freedom correspond to? This is physics after all! You've eliminated all of the excess mathematical baggage, and you still have TWO degrees of freedom - they must correspond to TWO different things!

However you point out that there is only one dof associated to the left-handed particle state and that a second dof actually comes from the right-handed antiparticle state.

I am trying to reconcile this with the above picture of a left-handed spinor simply being a two-component column vector that is an eigenstate of the helicity operator.

Until you brought it up, I had never thought about the antiparticle degrees of freedom, so I used to think that when we said that there are two fermionic dof in a chiral multiplet, those were the two dof of the left-handed two component spinor described above.

Now, if I want to include the CPT conjugate of the above, two component, spinor, I can imagine that what will happen is that the parameters of the CPT conjugate will be determined by the components of the left-handed particle so I could see how we could still end up with two dof even after including the CPT conjugate.

But it is still difficult for me to see that we can assign one dof to the left-handed particle state and one dof to the right-handed antiparticle state.

The only way I could imagine to make the connection with the way you explain it would be if there is a way to somehow do some type of linear transformation on the parameters of my left-handed two component particle Weyl spinor such that only one indepedent parameter ia actually left and that under CPT, this parameter is mapped to a totally independent parameter. I am not sure how to explain it clearly.

But the point is that my left-handed particle spinor seems to me to have two parameters so two dof. And I am trying to reconcile this with your explanation that the helicity=-1/2 state of the chiral multiplet is only one dof and accounts entirely for the particle state.

I'm not entirely sure I understand what you're saying, but just try to think of it in terms of the Fock space - there are 2 DOF in the Weyl spinor, so there are two physical states that can be produced:

$$\psi_{1}^\dag|0\rangle$$
$$\psi_{2}^\dag|0\rangle$$

These two states correspond to a particle with spin aligned with a given direction, and an antiparticle with spin anti-aligned with the same direction. The fact that the same Weyl spinor was able to make BOTH states once again involves CPT theorem - that you needed to be able to make them both!

What else can they be?! You have two DOF, so you can produce TWO states. You are saying that the "left handed particle has two DOF" - what are they physically?!

All this stuff about the LT of the fields is fine, but it's not telling you anything about the Fock space, except that it must be two-dimensional!

I will wait until I understand the dof of the Weyl spinor to get back to a question about this. I think of a Majorana spinor as being a four-component spinor which obeys a special condition, that the charge conjugation operator applied to it gives the spinor back. I had never thought aout it in terms of what type of mass term was written! I want to get back to this intriguing point later.

Our statements are isomorphic! However, I have been trained to think in terms of 2-component spinors, where the "C" operator you mention is simply unnecessary! The SM is a chiral theory, and therefore it is natural to always think in terms of Weyl spinors (even vectorlike theories like QCD or QED, there's no reason not to use Weyl spinors - the algebra is actually much easier, or so I think!). Your def of Majorana spinor is just "two Weyl spinors where the the two are equal to each other!" That's what "invariant under C" means in two-component language. As to which mass they have: that's the only physical distinction between "Majorana" and "Dirac" - again, in 2-component language, this is immediately obvious.

No-renormalization theorems are amazing. I read an article where theer was an interesting quote (I am not sure if it's Duff or Schwarz) saying something to the effect that having no renormalization is a theorist dream since it allows to obtain exact results from perturbative calculations!!

Yup, it's pretty impressive stuff.

Wow, I am very impressed that you remembered my connection with him!
yes, this is indeed a very good idea. I am a bit intimidated by him and I feel bad because I was going through a very difficult period in my life when I joined his group as a postdoc and I am sure I did not impress him very much. I feel like I need to get a few good papers out to prove myself a bit before contacting him, but that's a catch 22 since it's very hard to publish papers when having been out of the field for a while and having nobody at all to interact with!

Your suggestion is a very good one!

Well, I know nothing of your past other than what you told me when I was new to the forum, so I can't give you advice about reintroducing yourself to Cliff! But I can say that in my personal experience, he's always been very welcoming and happy to try and answer my questions, and I knew that you had worked with him before, and he's an expert in all the things you said! So I just threw his name out.

He/She must had said ''this guy is nuts!!!

Yes, it's a huge list and each of these topics could take years to master. I am not going in depth, obviously. What I meant is that I am trying to gather some basic background in each topic, just enough that I have some basic understanding of the concepts and of the calculations involved. It just feels sometimes that without a good background in susy, string (including branes), CFT and a few otherthings, it's almost impossible to understand anything at all that is being done. Most papers I read will quickly quote some results from these fields in the first few paragraphs of the paper. So it's frustrating not to be able to understand what is going on.

You (and your colleague) are completely right. It's just that it feels like everything is interconnected. Even if I start by focusing on one topic, it quickly feels like so many other things are required. A paper on brane worlds will refer to some result from brane stuff from string theory...to understand that stuff will require to understand BPS states and SUSY...and some other result will be a calculation done using CFT and so on.
The whole field has become monstrous!

You are completely right and I realize that my problem is hat I am trying to get up to speed all my myself so that I don't know what to set aside and what to focus on. I basically would need to do another PhD with an adviser guiding me....

All true! Ah, to be new to the field and want to learn everything about everything! I still have that problem: everyone's advice to me as a postdoc is to "stop doing that, pick something and master it!" I haven't done well by their advice...

Really!
At what level? I found teaching fun at first....but after a few years of full time teaching at a low level, I am going crazy I must admit. Teaching at a high level is great fun. And the years I put all research aside made me quite depressed.

Patrick

That's interesting. Well, this is another topic for another post, so let's not get sidetracked. I'll just answer your Q by saying I'm thinking of applying for teaching colleges where I can do research as well. I really like teaching (that's why I joined this forum) and as a postdoc I miss it.

OK, let's leave it at that.

Homework Helper
Gold Member
not at all!

but ask yourself: WHAT do those two degrees of freedom correspond to? This is physics after all! You've eliminated all of the excess mathematical baggage, and you still have TWO degrees of freedom - they must correspond to TWO different things!

I'm not entirely sure I understand what you're saying, but just try to think of it in terms of the Fock space - there are 2 DOF in the Weyl spinor, so there are two physical states that can be produced:

$$\psi_{1}^\dag|0\rangle$$
$$\psi_{2}^\dag|0\rangle$$

These two states correspond to a particle with spin aligned with a given direction, and an antiparticle with spin anti-aligned with the same direction. The fact that the same Weyl spinor was able to make BOTH states once again involves CPT theorem - that you needed to be able to make them both!

Ok, that makes sense. But that shows that in order to really understand those dof, one must in fact focus on the quantum fields and see the properties of the states created by the quantized mode operators. I was trying *not* to think in terms of QFT but in terms of QM states and what I was seeing was a two component wavefunctions obeying the Dirac equation and which is an eigenstate of the helicity operator with an eigenvalue of minus one. And which transforms in the way I described.

Because I was thinking of a single particle state, it suggested to me that we were dealing with two dof for the single left-handed particle. I did not see how CPT was implemented at that stage, only Lorentz invariance.

I feel that there is an important and nontrivial piece of the picture that I was missing. I was trying to take seriously your suggestion to think in terms of QM, not QFT. In that case I simply had a two component wavefunction satisfying a (covariant) ''Schrodinger equation''. I did not even think about considerations of CPT.

But we know that a relativistic interpretation of a single particle wavefunction is inconsistent and that leads to QFT and so on. So now treating the two components Weyl spinor as a quantum field, one introduces two types of creation operators and those correspond to the two states you described.

I am not sure if that makes sense to you but this is how I understand the situation now.

What else can they be?! You have two DOF, so you can produce TWO states. You are saying that the "left handed particle has two DOF" - what are they physically?!

All this stuff about the LT of the fields is fine, but it's not telling you anything about the Fock space, except that it must be two-dimensional!

I understand. And the second quantization language had to be used in order to understand what the creation operators associated to these dof were creating.
Our statements are isomorphic! However, I have been trained to think in terms of 2-component spinors, where the "C" operator you mention is simply unnecessary! The SM is a chiral theory, and therefore it is natural to always think in terms of Weyl spinors (even vectorlike theories like QCD or QED, there's no reason not to use Weyl spinors - the algebra is actually much easier, or so I think!). Your def of Majorana spinor is just "two Weyl spinors where the the two are equal to each other!" That's what "invariant under C" means in two-component language. As to which mass they have: that's the only physical distinction between "Majorana" and "Dirac" - again, in 2-component language, this is immediately obvious.
That makes sense.

But I can say that in my personal experience, he's always been very welcoming and happy to try and answer my questions, and I knew that you had worked with him before, and he's an expert in all the things you said! So I just threw his name out.
And I think it's an excellent suggestion. He is indeed a very nice and approachable guy and incredibly smart.

All true! Ah, to be new to the field and want to learn everything about everything! I still have that problem: everyone's advice to me as a postdoc is to "stop doing that, pick something and master it!" I haven't done well by their advice...

That's interesting. Well, this is another topic for another post, so let's not get sidetracked. I'll just answer your Q by saying I'm thinking of applying for teaching colleges where I can do research as well. I really like teaching (that's why I joined this forum) and as a postdoc I miss it.

OK, let's leave it at that.
I understand the feeling, I felt exatly the same way when I was doing my postdoc. There are places were research is appreciated and those places offer an ideal mix of teaching and research.

Next topics will be massive states in 4D and then other dimensions. I will get back with more questions soon, after going back to what you have already covered on those topics.

Thanks again!

blechman
Ok, that makes sense. But that shows that in order to really understand those dof, one must in fact focus on the quantum fields and see the properties of the states created by the quantized mode operators. I was trying *not* to think in terms of QFT but in terms of QM states and what I was seeing was a two component wavefunctions obeying the Dirac equation and which is an eigenstate of the helicity operator with an eigenvalue of minus one. And which transforms in the way I described.

Because I was thinking of a single particle state, it suggested to me that we were dealing with two dof for the single left-handed particle. I did not see how CPT was implemented at that stage, only Lorentz invariance.

I feel that there is an important and nontrivial piece of the picture that I was missing. I was trying to take seriously your suggestion to think in terms of QM, not QFT. In that case I simply had a two component wavefunction satisfying a (covariant) ''Schrodinger equation''. I did not even think about considerations of CPT.

But we know that a relativistic interpretation of a single particle wavefunction is inconsistent and that leads to QFT and so on. So now treating the two components Weyl spinor as a quantum field, one introduces two types of creation operators and those correspond to the two states you described.

I am not sure if that makes sense to you but this is how I understand the situation now.

....

I understand. And the second quantization language had to be used in order to understand what the creation operators associated to these dof were creating.

I think so. Second quantization definitely plays a role, and when I said "QM" I implicitly meant "relativistic QM". I'm not used to making such a distinction between QM and QFT and I might have said some misleading things - sorry about that. But I think you have the idea now.

I understand the feeling, I felt exatly the same way when I was doing my postdoc. There are places were research is appreciated and those places offer an ideal mix of teaching and research.

Yeah, I was thinking of applying at such a place. They're very competitive jobs to get. But in any event, I won't get sidetracked with this for now.

Next topics will be massive states in 4D and then other dimensions. I will get back with more questions soon, after going back to what you have already covered on those topics.

Thanks again!

Keep it coming! I'll be going on a short, well-deserved vacation for the weekend, but I'll be back next week, so you might have to wait for a response. I'm just giving you advanced warning so you know I'm not trying to ignore you.