You must have seen this table. What we see around seems to be made out of atoms. They have electrons \(e^-\) orbiting around nuclei and nuclei seem to be composed of protons \(p\) and neutrons \(n\). For over 40 years, we have known that protons and neutrons aren't elementary. Each of them is composed (mostly) of three quarks, \(uud\) and \(udd\) where \(u,d\) stand for the up-quark and down-quark, respectively.

We also experience light and electromagnetic waves which may be shown to be composed of elementary quanta, the photons \(\gamma\). For quite some time, people could have thought that everything was made out of \(e^-,u,d,\gamma\). Well, once the quarks were known, physics has had learned about other particles as well, but there exists an alternative history in which this subtlety didn't arise. ;-)

However, in the late 1920s, Paul Dirac realized that each particle had an antiparticle. Most famously, the electron's antiparticle is called the positron (the only commonly used "irregular" name of the sort, one that replaces the "antielectron").

Most particle species have an antiparticle that differs from them. Electrically neutral elementary particles tend to have an antiparticle that is identical to them. This is surely the case of the photon \(\gamma\), which is therefore the same thing as the antiphoton, the newly discovered Higgs boson \(h\), and possibly also for the neutrinos \(\nu\) if their masses are "Majorana masses" (but we must realize that the neutrinos' being left-handed or right-handed effectively distinguishes two "species" in a similar way as if they were the particle and the antiparticle).

The neutron \(n=udd\) isn't an elementary particle so it is very different from its antiparticle \(\bar n=\bar u \bar d\bar d\).

During the 20th century, people realized that there existed heavier brothers of the electron which have the same charges but are different and heavier. They're called the muon \(\mu^-\) and the tau \(\tau^-\). Each of them has a distinct antiparticle and a neutrino from the list \(\nu_e,\nu_\mu,\nu_\tau\).

In total, we have these six leptons,\[

e^-,\mu^-,\tau^-; \quad \nu_e, \nu_\mu, \nu_\tau.

\] You may imagine that the antiparticle to each of these 6 particles is a different object such as \(\mu^+\) or \(\bar \nu_\mu\) although for neutrinos, they could be the same. All these particles carry spin \(j=1/2\) and are described by the Dirac equation or a "halved" version of it, the Weyl equation or the Majorana equation.

Aside from these six particles, there are also six types of quarks – we call them flavors. The previously mentioned up- and down-quarks are just the first generation and three generations are known:\[

u,c,t; \quad d,s,b

\] The verbal names are up, charm, top (=truth); down, strange, bottom (=beauty). The first three have the same charges as the up-quark, the last three resemble the down-quark. The antiparticle to each of the six quarks has to be a new particle (denoted by a bar) because their charges differ.

This completes the list of elementary fermions; all of them have \(j=1/2\) and they're divided to leptons (first six) and quarks (last six). The only other allowed spin of an elementary fermion is \(j=3/2\) and if a particle has it, it has to be a gravitino \(\tilde G\), a superpartner of the graviton.

Now the bosons. The spin \(j=1\) photon has two or three heavy brothers, \(Z^0, W^+, W^-\) where the charged W-bosons are antiparticles to one another and others are self-adjoint in this antiparticle association. All these four particles are gauge bosons of the \(SU(2)_W\times U(1)_Y\) gauge group of the electroweak theory.

The photon is the only massless combination because the Higgs vev is only neutral under the corresponding charge, the electric charge \(Q\), and this fact has implications: electromagnetism is the only long-range force (aside from gravity mediated by the equally massless graviton). The other forces in the electroweak group are short-range forces and only operate "inside the nuclei", so to say. They're mostly known as the causes of the radioactive decay of the neutron,\[

n\to p+ e^- +\bar\nu_e,

\] and the corresponding \(\beta\)-decay of nuclei (composites of the decaying neutron, perhaps other neutrons, and protons).

Aside from the four electroweak gauge bosons with \(j=1\), there are eight components of the gluon \(g\) which is the gauge boson of the 8-dimensional group \(SU(3)_c\) where \(c\) stands for "color". This gauge group is behind QCD and makes the quarks hold together inside protons and neutrons (and the residual force between these color-neutral nucleons i.e. protons or neutrons is still enough to hold the nucleons together in the nuclei).

Gluons are not really massive in any rigorous enough sense but they are charged under the color group \(SU(3)_c\) which makes them interact with other gluons and they are actually subject to "confinement", too (just like the quarks). So you can't isolate a single gluon (just like you can't isolate a single quark) and the force is effectively a short-range force, too.

This completes the known \(j=1\) bosons. The only known \(j=0\) (=scalar) boson is the Higgs boson discovered in 2012. The only other allowed value of the spin of the elementary particle is \(j=2\) and only the graviton is allowed to have such a high value of the spin.

**Now, add supersymmetry**

Supersymmetry is a symmetry generated by a fermionic, spin \(j=1/2\) generator \(Q_\alpha\) where \(\alpha\) is a spinor index (and the Hermitian conjugate generator \(\bar Q_{\bar \alpha}\)). It commutes with the Hamiltonian (because it is a symmetry) and due to the rules for the addition of the angular momentum, it inevitably transforms a particle of spin \(j\) to a particle of spin \(j\pm 1/2\), the "superpartner". The whole collection of component particles including the superpartners is known as a "supermultiplet", a representation of the group/algebra that we call the "supersymmetry algebra". (The fields producing the particles in the supermultiplet may also be written as "superfields" i.e. fields that depend on the known \(3+1\) coordinates as well as some new Grassmann/fermionic coordinates of the "superspace".)

A negative value of \(j\) is impossible so in practice, we will only need the supermultiplet containing \(j=0\) and \(j=1/2\), a scalar particle and a fermion, and one more mentioned in the next paragraph. The \(j=1/2\) fermion is conveniently thought of as a Majorana ("real") fermion for neutral enough particles or, more precisely, a Weyl ("chiral" but complex) fermion for the charged ones. This supermultiplet is known as the "chiral multiplet" because the fermion in them must be chiral. Such multiplets are relevant for the leptons and quarks of the Standard Model (adding \(j=0\) sleptons and squarks) as well as the Higgs field (in the opposite way, adding \(j=1/2\) higgsinos).

The other allowed supermultiplet is called a "vector multiplet" and it contains \(j=1\) bosons and their \(j=1/2\) fermionic superpartners. So the photon \(\gamma\), the W-bosons, the Z-boson and the gluon have new \(j=1/2\) superpartners, the gauginos.

The combination of spins \(j=1\) and \(j=3/2\) isn't possible because the only way to remove the negative-norm components in the \(j=3/2\) field is a local supersymmetry algebra and it inevitably closes on the spacetime translations, i.e. gravity, so the only allowed multiplet with a \(j=3/2\) fermionic field is the gravitational multiplet which contains the \(j=3/2\) gravitino \(\tilde G\) and the \(j=2\) gravitons \(g_{\mu\nu}\). The theories with this set of particles are known as supergravity (or SUGRA) or have SUGRA at low energies (the case of superstring theory). The spin \(j\geq 5/2\) is already too high for elementary particles.

So what are the particles of the Minimal Supersymmetric Standard Model after the doubling?

All superpartners of ordinary particles are denoted by a tilde, e.g. \(\tilde t\). The new scalar (\(j=0\)) partners of the known \(j=1/2\) fermions are called by a word with the s- prefix ("s" stands for "scalar" – or "super", if you wish). So we have sleptons and squarks such as sups, stops, sbottoms, smuons, and sneutrinos, among others.

The new \(j=1/2\) superpartner of the \(j=0\) Higgs boson is known as the higgsino. Higgs' name isn't capitalized here because Higgs is remolded in this word and lucheon meat just doesn't need capitalization. The "-ino" suffix is used universally for all fermionic partners of known bosons.

That's also true for the \(j=3/2\) gravitino superpartner \(\tilde G\) of the \(j=2\) graviton as well as the superpartners of the \(j=1\) gauge bosons, the \(j=1/2\) gauginos. They are called the photino, zino, wino, and gluino. However, this simple "doubling" has two important technical subtleties that require a somewhat more complex discussion:

- need to extend the Higgs sector
- need to rediagonalize the mass matrices

So instead of one complex doublet (=4 real components), we need two complex doublets (=8 real components in the scalar fields) plus their fermionic superpartners. In the normal Standard Model, the counting of physical Higgs polarizations goes like \(4-3=1\) because three of the four real components are "eaten" by the three massive gauge bosons (\(W^-,Z^0,W^+\)) which need to extend their number of polarizations from \(2\) to \(3\) once they become massive.

The analogous counting of the scalar fields in the Minimal Supersymmetric Standard Model (MSSM) is \(2\times 4 - 3 = 5\) because there are two doublets but still just three massive gauge bosons which is why there are five Higgs scalar bosons in the MSSM. Two of them are charged and antiparticles to one another, \(H^\pm\), and three of them are neutral (and self-adjoint when it comes to particle-antiparticle pairings). The neutral ones are \(h,H,A\) where the first one is meant to be the known Higgs boson, the second one is probably heavier (but there's a possibility that the yet-to-be-discovered Higgs is actually lighter than the known one), and \(A\) is the only CP-odd Higgs boson among the three (the CP-conjugation maps charged particles to different, differently charged particles so charged particles can be neither CP-odd nor CP-even).

**Now, the second point: the need to rediagonalize the masses.**

The energy is an operator, a matrix in quantum mechanics, not a scalar. So it's also true for the energy of an object in its rest frame, also known as the mass. So the masses are matrices that may have off-diagonal elements. The observed masses of particles are the eigenvalues of these matrices.

So we need to diagonalize these matrices; the off-diagonal element imply that the original simple-minded superpartners such as the higgsino and the photino are no longer mass eigenstates (their mass is no longer well-defined). Only some linear superpositions remain mass eigenstates.

The off-diagonal mixing between two previous ket vectors (e.g. photino, higgsino) exists whenever these two ket vectors have the same charges (the charge conservation [and angular momentum conservation] prohibits mixing i.e. off-diagonal elements between two particles with different charges [or spins]).

It means that in principle, we have to throw all the fields with the same charges (and same spins) to a "melting pot" and choose the same number of mass eigenstates from the melting pot. For example, the three up-type squarks, namely sup, scharm, and stop, have to be thrown into such a melting pot and three eigenstates have to be extracted. In this particular case, the mixing has to be small because if SUSY allowed a strong mixing, it would generate too strong "flavor changing processes" which are experimentally prohibited. Similar comments apply to all quarks and leptons.

However, the left-handed and right-handed quark fields have two independent complex scalar superpartners and those can mix with each other. For example, one of the stops may be lighter than the other. The lighter one may be the left-handed one \(\tilde t_L\), the right-handed one \(\tilde t_R\), or some superposition of them known as \(\tilde t_1\). The \(\tilde t_2\) denotes the heavier mass eigenstate.

It's pretty much the case that the mixing between sup, scharm, and stop is neglected, much like for the sdown, sstrange, and sbottom; selectron, smuon, stau; and the three sneutrinos. So the terminology for the superparticles still reflects the simple doubling.

However, the mixing is allowed (well, encouraged) for the gauginos and higgsinos. The gluinos \(\tilde g\) are the only elementary fermions in the adjoint of \(SU(3)_c\) so they can't mix with anyone. The terminology "gluino" is therefore fine.

On the other hand, the higgsinos and the photinos, winos, and zinos have to be thrown to the two melting pots. Some of these particles carry \(Q=0\) and they produce new \(j=1/2\) Majorana fermions called "neutralinos". Some of them are charged, \(Q=\pm 1\), and they produce charginos.

When you count the number of components – it's useful to count the original fields and ignore the fact that some of the polarizations of the Higgs doublet are "eaten" by the gauge bosons – you find out that there are four neutralinos and two charginos (plus their two antiparticles). More precisely, the rediagonalization yields four mass eigenvalues for neutralinos (which are some linear superpositions of the neutral higgsinos, photino, and zino) and two distinct mass eigenvalues for the charginos (which are superpositions of charged higgsinos and winos). Note that if you count a complex field and its complex conjugate as having the same number of components as a pair of real fields, the number of fields in the MSSM neutralinos is the same as the number of fields in the MSSM charginos (this fact is clear for the Higgs doublets because 1/2 of the doublet is neutral and 1/2 of it is charged; for the 4-dimensional \(SU(2)\times U(1)\) group, there are 2 "neutral" generators and 2 "charged ones"). The charginos only have two and not four mass eigenvalues because the eigenvalues have to be paired due to the CPT symmetry.

For those reasons, you won't really hear particle physicists talking about photinos and zinos too often. Instead, they will usually start with the original basis of the gauge fields, the W-field for \(SU(2)_c\) and the B-field for the hypercharge \(U(1)_Y\), and talk about the winos (including the neutral one) and the bino. Even more generally, they usually throw the higgsinos to the mix and replace all the zinos \(\tilde Z\), photinos \(\tilde \gamma\), winos \(\tilde W^\pm\), and higgsinos \(\tilde h\) etc. by "electroweakinos" i.e. neutralinos \(\tilde\chi^0_{1,2,3,4}\) and charginos \(\tilde\chi^\pm_{1,2}\). The superscript denotes the electric charge and the subscript labels the eigenstates starting from the lightest ones.

**Dark matter candidate**

The lightest superpartner (lightest among the particle species that are implied by SUSY but not yet known) is interesting because it has good enough reasons to be stable (or long-lived). The reason is (exact or approximate) R-parity conservation where the R-parity is \(R=\pm 1\) and it is \(R=+1\) for the known particles and \(R=-1\) for their superpartners. If the R-parity is conserved, the number of superpartners has to be conserved modulo 2, so they can't "completely" decay away. The lightest one is guaranteed to be stable because it has nothing to decay into (without violating the conservation of \(R\)).

*Only the black girls at the end seemed to be really afraid of the flying dark matter.*

This stable particle may therefore be the dark matter candidate – some particle species that has survived for billions of years. If you want to make it dark i.e. not interacting with the light (photons, electromagnetic field), it should be electrically neutral, most likely the lightest neutralino \(\tilde\chi^0_1\).

Well, I still haven't been clearly told why the dark matter can't be charged, e.g. charginos, creating "very heavy hydrogen atoms" (hydrogen atoms with an almost ordinary spectrum but with a very heavy nucleon) which could produce the additional mass needed to solve the dark-matter mystery without having any new matter that is "dark".

At any rate, the most widespread WIMP (weakly interacting massive particle, the dominant paradigm for dark matter) in the literature is the LSP and the most popular LSP is the neutralino \(\tilde \chi^0_1\). This has a nonzero chance to be observed tomorrow, too.

*Our dark matter poll as seen 27.2 hours before the October 30th announcement: out of 238 respondents, 59% say that nothing will be seen, 27% think that a light dark matter particle will be confirmed, 8% expect a heavy dark matter particle, and 7% suggest something else.*

However, the dark matter particle may also be the gravitino – in that case, the R-parity doesn't have to be conserved because the gravitinos' annihilation is very slow due to the weakness of gravity even if the R-parity is violated. In that case, we also need to talk about the NLSP, the next-to-lightest superpartner.

And the dark matter particle may be a superpartner to a currently unknown, new particle, too – the singlino \(\tilde s\) where both \(s\) and \(\tilde s\) are experimentally unknown at the present.

**All known fermions on a tie**

It's generally believed that the equations of a theory of everything should be printed on T-shirts; some top physicists have even applied the rule to a theory of nearly everything. The analogous statement about the known fermions is that they should be printed on ties. I got these two ties from a very kind woman, Jana S., and they seem to include all the known/visible fermions:

It's the leptons. You might be worried that I will have to throw the ties to the trash bin tomorrow in the case that they announce the discovery of a dark matter particle in South Dakota. But this ain't the case because the ties have a dark side, too:

Well, we were using the notation \(\tilde \chi\) with the tilde but it's good enough.

Experts will recognize that the knot is the Manhattan tie knot – I accidentally picked it from an iOS app after the app was nearly destroyed by an update that has replaced one knot by about \(10^{500}\) options and I couldn't remember what was the old one. Surprisingly for some, it still worked equally well after the update and I even wore the tie & knot above on TV last month. ;-)

Bosons should be printed on the underwear but I will leave the job of publicizing these things to others. Especially the location of the graviton might have an interesting explanation by a female physicist.

What's the trick with the Higgs not being it's own antiparticle in the early universe and higgsogenisis?

ReplyDeletehttp://www.nature.com/news/higgsogenesis-proposed-to-explain-dark-matter-1.13883

Well, try to see the paper

ReplyDeletehttp://arxiv.org/abs/arXiv:1304.3464

Some new fields are added, the distinction Higgs-anti-Higgs surely can't exist in the Standard Model. I don't understand in what sense the asymmetry between H and H^dagger may arise, anyway. Asymmetry in what symmetry? Why would one use the "Higgs field" for anything else than the absolute value of the Higgs doublet which is real and therefore inevitably identifying particles and antiparticles?

The new discrete symmetry that's broken is supposed to be good enough for baryogenesis in that paper but I don't understand the motivation of it and I don't understand the terminological spin they use, either.

At any rate, you shouldn't allow a strange 4-citation paper blur the indisputable fact that in the Standard Model or any sufficiently peaceful extension of it, the electrically color neutral Higgses are always identical to their antiparticles.

Excellent article Lubos. Question: If there are photinos and winos, will they modify electromagnetic and weak processes at high energies? My understanding is that other than spin they would have couplings to other particles similar to what photons and w bosons have. Is this right? But then, is it true that the force carriers (as gauge bosons) must have integral spin only? Do you think it is possible to see these modifications at LHC energy or they are very complicated anyway, so it may be difficult to see?

ReplyDeleteDear Kashyap, all particles have some consequences - because of the more or less complicated Feynman diagrams with these particles in the role of virtual particles.

ReplyDeleteBut you are clearly imagining the similarity between the photinos (neutralinos) etc. and photons to be closer than it is.

Don't forget that neutralinos are *fermions*, unlike photons which are *bosons*, so these neutralinos couple to a list of other particles in the vertex that contain an odd number of additional fermions. So they can't be coupled as photino-electron-positron because such a vertex would contain an odd number of fermionic factors and would be fermionic (and would have half-integral spin) which violates the angular momentum (and Grassmann parity) conservation law.

So the couplings are different than you imagine. If there's a coupling Z-neutrino-antineutrino, then there is a coupling zino-neutrino-antisneutrino - note that one of the neutrinos is replaced by its superpartner, too.

If such a zino flows in, then it changes the neutrino to its superpartner in the vertex and if you only want to get neutrinos back, there has to be another vertex that does the opposite reversion again.

Also, zinos and higgsinos etc. can't have vevs or other macroscopic values of the classical fields; fermionic fields classically have vevs equal to zero at all times. So yes, force carriers have to be bosons i.e. have integer-valued spin. There isn't any force transmitted by a fermionic field because this field, when multiplying another field, creates yet another field (with the opposite statistics) so you can't get a nonzero "charge" (eigenvalue).

Wait, were you really on TV last month or was that a joke I failed to appreciate?

ReplyDeleteNo, there is no joke here. It was the IPCC report on Sept 27th. At 10 pm, it was the first news of the news channel of the Czech Public TV, Čt 24, and I was there remotely from videochat in the Pilsen TV studios to counterbalance what Bedřich Moldan, a fomer minister of environment, and Radim Tolasz, the current boss of the Czech IPCC delegation, had to say.

ReplyDeleteI must mention that despite Moldan´s membership in the center-right Klaus-founded ODS, he acted like a much more mindless alarmist than Tolasz who may be said to be apolitical.

Lubos,

ReplyDeleteThe paragraph that begins "The analogous counting" says there are "three doublets", don't you mean two?

Nice didactic blog post, Lubos. The level is pitched just about right (at least for me).

ReplyDeleteYup, it was a mistake, fixed, thanks!

ReplyDeleteHa ha...Are you trying to be like Matt Strassler? Don't even attempt it. He's a much better blogger and explainer than you. AND his blog entries are much shorter and to the point.

ReplyDeleteAnd he doesn't insult people nor call people names, unlike you.

ReplyDeleteAnd he already has articles about topics such as this. Just post a link to HIS blog and call it a day.

ReplyDeleteI applaud your derivation but fault your premise. LHC observes no string/brane exotica, squarks, sleptons, bosinos, leptoquarks, lazy photons, WIMPs, colorons, supersymmetry exotica, extra-dimensions, magnetic monopoles, mini-black holes, Randall-Sundrum 5-D phenomena (gravitons, K-K gluons, etc.), ADS/CFT duality evidence, fractionally charged particles... Sterile neutrinos and the "see-saw" mechanism rationalizing divergent neutrino-antineutrino decay paths are obscene.

ReplyDeleteSUSY and MSSM are unphysical. Zero CERN/CAST solar axion detection. Zero Super-Kamiokande proton decay. Quantum spacetime lacks exact vacuum isotropy toward matter,

arXiv:1109.1963. Dark matter curve-fits the Tully-Fisher relationship universally modeled by MOND's Milgrom acceleration. Milgrom is Noetherian leakage of angular momentum conservation from trace chiral vacuum anisotropy toward matter, validated by 90-day geometric Eötvös experiments. A non-null net output ends string/M-theory and quantum gravitations postulating the Equivalence Principle. GR superset ECKS gravitation is validated.Perform a geometric Eötvös experiment, left-handed versus right-handed single crystal alpha-quartz test masses. If a net null signal obtains, I will abase myself with public apologies. If a net non-zero signal obtains, you publish voluminously, having had years head start for deriving quantum gravitation ansätze with trace chiral spacetime torsion.

You know. My problem overall is that at every juncture there needs to be yet another even more involved explanation for yet another unexpected/unresolved event. This is becoming almost exponentially complex to explain the theory, which is the exact opposite from how a theory should be. This whole thing screams for some kind of anti-Ockham meme.

ReplyDeleteBut not just this here, the whole enterprise towards a TOE is just like walking uphill through shifting sand.

With each step it gets harder, and you make less progress

Wicked humor. Love it.

ReplyDeleteI hear this often from the laymen but these comments make no sense to me. A theory should be as simple as possible but not simpler. If one discovers the dark matter - a deviation from simpler laws, he has to add the effect to his theory. If he discovers a particular new particle, the theory has to agree with its existence, too. This is what science dictates without any tolerance. In comparison, Occam's razor is just a vague unreliable medieval superstition.

ReplyDeleteMoreover, it's not true that things are getting more complex. Things have been getting more unified in the recent centuries and decades. Millions of compounds, material, and species have been reduced to just a dozen or two of elementary particles, with a few terms, and they're organized into various families making the existence of some of them an inevitable consequence of the existence of others. String theory probably makes the existence of all of them inevitable. But even if something isn't inevitable and derivable out of nothing, it doesn't mean that it doesn't exist.

A certain amount of hubris isn't strange to humankind but one shouldn't exaggerate. Talking down to people isn't a sign of intelligence but of arrogance.There is no way of ascertaining there is a connection between theory and 'reality'. It's just something 'we' concoct so things we observe with our limited timespan seem to overlay events in a way it makes sense. It might just as well be the universe as we observe it ceased to exist already but the signal just can't reach us since that would be a paradox. To my mind our lack of oversight in the random flow of energy that clots together to form this particular universe and to us seem to last in a 'time' is anything but just that. A fleeting happenstance of where the laws you observe seem to be in place and are gone the same instant in reality.

ReplyDeleteA fruitfly must think it's life lasts long a time too, where untold adventures take place.

I don't know what to do with someone who says "There is no way of ascertaining there is a connection between theory and 'reality'."

ReplyDeleteThe entities you claim not to exist are called evidence and rational reasoning. Saying that those things don't exist may be classified as arrogance or non-arrogance or whatever irrelevant labels you discuss but what it primarily and indisputably is is stupidity and a denial of reality.

Sorry, we clearly can't agree about anything related to science - or any rational thinking, for that matter - here or elsewhere.

I guess not. But you sure have long toes.

ReplyDeleteLumo, is there a relation between electromagnetism and spin or angular momentum? Is it fundamental for electromagnetism?

ReplyDeleteYup, there are many relationships like that, but there's one of your kind that is actually pretty deep.

ReplyDeleteElectromagnetism may be derived from a "gauge symmetry" and "gauge symmetry" is something one has to demand to exist for particles and fields - photons and electromagnetic fields - that carry spin (intrinsic angular momentum).

So to some extent, all properties of electromagnetism follow from the photon's having a nonzero spin, or from the electromagnetic waves' carrying angular momentum (or from their being transverse).

Reading tons of comments by all these layman but sometimes physicists in fields far away from HEP (th), who simply dont get it or are even outright and intentionally trolling, is so stupid and tiresome :-(

ReplyDeleteAs if wrong assertions would become true by repeating them zillions of times everywhere ...

Is this troll not a perfect black list candidate :-)

ReplyDeletelet's say "invariance to the redundancies of the formulation"...

ReplyDeleteCould you please tell us how you generate such random nonslense? It would be a good tool to have.

ReplyDeleteLOL :-D

ReplyDeleteIt's funny how $\Pi^\mu$ is the embedding function onto superspace given by $\Pi _\alpha ^\mu =.

ReplyDeleteThe difference seems to have to learn sThe difference seems to have to learn some Matrix Calculusome Matrix Calculus

ReplyDelete