Joël Scherk (not Schrek, the ogre) was a French theoretical physicist with a tragic death. Perhaps his contributions were short, but they were important in the fields of supersymmetry (SUSY), supergravity (SUGRA) and superstrings. Moreover, he also introduced some crazy ideas related to these things I am going to discuss. This blog post is essentially a refurbished reedition of his famous talk giving an overview about SUGRA, SUSY and antigravity. I have added some extra stuff by my own, and I plan to update this post in the future with more topics involving the phenomenon of antigravity that Scherk discovered long ago. Enjoy it!

First of all, even when I have not (not yet) dedicated a thread to SUSY or SUGRA, I will be discussing some of his ideas with respect to them.

What is SUSY? SUSY is just a particular type of symmetry or transformation group. Generally speaking, SUSY is defined in terms of transformations which leave invariant certain action AND transforms fermions (or fermionic fields to be more precise) into bosons and viceversa. I have discussed group theory in this blog (elementary group theory) and Lie algebras too. There is a generalization of Lie algebras called Lie superalgebras, also known as graded Lie algebras. Graded Lie algebras (GLA or superalgebras) are the elementary mathematical structures and tools used in SUSY/SUGRA. GLAs used in SUSY and SUGRA models, and their mathematical representations analogue to those in Lie algebras, will be reviewed in this post, based on a famous Scherk’s talk…I am going to explain you some bits and glimpses of the (extended) Poincaré GLA, the (extended) Poincaré de Sitter GLA, and the conformal GLA. I will be also introducing you the N=8 SUGRA and its relevance to physics.


The mere existence of fermions can be traced back to the early 20th century. Around 1925, physicists found the strange and weird character of the electron and similar particles. “Why fermions?” Louis de Broglie asked that question, even when the Pauli exclusion principle was formulated. So we are in the situation where Nature chose the electron to be a fermion (proton an neutrons are fermions too). From the experimental aside, we can not deny but accept the existence of fermions: spin 1/2 particles! What about spin s/2 particles, with odd s greater than 1? We will learn about this just a little…Focusing on the 1/2 case…Quarks are also fermions! Fundamental forces, from the quantum viewpoint, are mediated by integer spin particles (gluons, photons, W and Z bosons and now, it seems, Higgs bosons too). We have also the hypothetical gravitons from the gravitational theories like General Relativity and extensions. Thus,  integer spin particles and its existence is out from any reasonable doubt. However, fermions are just a bit more mysterious. Poor mathematicians and mathematical physicists use to ask and wonder why fermions are relevant.

Simple answer: “There are fermions in Nature because we observe them”

Uhlenbeck and Gouldsmith told us in 1925 that the spin of the electron is 1/2, no matter how weird it is. It is an experimental fact! Furthermore, the spin-statistics theorem tells us that if we have spin 1/2 particles or fields (Dirac fields for simplicity at the moment), these should be quantized through Fermi-Dirac distributions/statistics, while if one deals with integer spin fields, their corresponding distribution is that of Bose-Einstein type. In summary, there are boson and fermions in Nature. The Standard Model (or Standard Theory) says that force carriers are boson fields, and that matter fields are fermion fields. This reasoning is, perhaps, not simpler that saying “A rose is a rose is a rose” (Gertrude Stein).

Elaborated answer: There is a deeper reason of spin 1/2 existence. From a mathematical viewpoint, it is ultimately related to the rotation group SO(3), and the Poincaré group. The SO(3) group is NOT simply connected and its universal covering group is SU(2). SO(3) admits integer representations only, but SU(2) admits 1/2 integer as well as integer representations. So from the SU(2) viewpoint, Nature has realized both integer and 1/2 integer representations. SO(3) is, if you want, more boring than SU(2). This second answer, however, leaves one with an unpleasant feeling, as one may wonder if mathematical physicists would have thought about SU(2) if Nature had chosen a world wher only gluons, photons, W and Z bosons, Higgses and gravitons existed, or if it had played the trick to provide us with fundamental (composite?) spin 0 scalar particles (higgses) and quarks (which is another option!).

Nature said, indeed, that there are Fermions, quarks and leptons, with J=1/2; there are also Bosons, gravity (hypothetical gravitons) with J=2 particles, electroweak particles (massless photons, and massive W,Z bosons, massive Higgses) with spin J=0,1, and strong interactions having gluons with J=1. Mathematical physicists (M\Lambda \Phi physicists) know that J(electron)=1/2, that electrons from spin-statistics imply the existence of Fermions while interactions imply the existence of (gauge/gauge like) compensating fields with J=0,1,2. Fermions (F) do exist. Why F? The response involves representations of the Poincaré group: SO(3) covering group is SU(2), which admits 1/2 spin fields (and other exotica I am not going to discuss here today). Math is OK, but why is realized by Nature? Not simple answer to this does exist. Would you search for some 1/2 piece of art? It does NOT exist spin 1/2 “art”. Metaphysical answers? The western way offers no simple explanation! The M\Lambda \Phi physicist turns towards Western rationalistic enlightenment without concrete solutions. You would only find unsatisfactory theological reasons and you will turn towards Eastern religions. The East gives you a more cool option, namely that a universal principle of harmony and dynamics balances the whole Universe (cf. the Greek idea of arche/arkhe). Chinese people call this Taiji principle the Tao. Don’t worry! I am not going to convert this article into the Tao of Physics ;). Tao refers to the interplay and inerfusion of pairs of opposite “elements” of fundamental character, dubbed Yin and Yang. The Yin refers to the female, dark , cold , soft, negative principle while the Yang (please, I am not the one who said all this even if you find it machist!) refers to the male, bright, warm, hard and positive principle/element (the Universe is mainly dark/female thought ;)).

Scherk tried to temp us with this analogy: one can associate the Yin principle with the Bose principle, that allow interpenetrability, and the Yang principle with the Fermi principle, which doesn’t. It is hard to be dogmatic on this scheme since the darkness/brightness would associate the Bose principle with the Yang and the Fermi principle with the Yin. Since the the light is known to be light/bright and not dark (unless dark photons exist), and the (SM) photon is a boson. More stupid stuff to waste time with Yin/Yang analogies in particle physics: little circles of black within white and vice versa signify that Yin bound states (stable or metastable) of the Yang principle exist and vice versa. This is known to be true as bound states of fermions (e.g. quarks) produce bosons, while the less obvious converse result is true as we know or learn about the study of classical magnetic monopole solutions. This complementarity extends in taoist thinking no only to bosons and fermions, but to many other subjects such as the World as an Object and the Ego as a Subject, which are in this way of seeing not only complementary but also interchangeable. Object/Subject complementarity leads to the cosmologist’s viewpoint (anthropocentrism, if you wish): the Universe or World is as it is in particular state because it is populated by conscious human observers (conscious E.T. beings or even conscious A.I. will also apply). The existence of life and consciousness puts some limitations on various purely physical constants such as gravitational/electromagnetic/weak/strong coupling constants and so on. This class of anthropic idea is not very popular or “well seen” between scientists. Applied to the Fermi/Bose problem, if Nature had decided for instance to build up us with J=0 leptons and J=1/2 quarks only, the nuclei would exist, and so would the atom in a modified way though. However, every atom would collapse under the gravitational strength into one giant molecule at the Earth’s center.

This last answer is also unpleasant. We turn into a more modern view of this problem. It is dominated by SUSY, SUGRA and superstring/M-theory. Local SUSY implies SUGRA. And it necessarily implies a complementarity between Bose (J=0,1,2) particles/quanta and Fermi (J=1/2,3/2) particles/states and their transmutability into each other through SUSY transformations, at least from the pure mathematical viewpoint! SUSY is equal to Fermi-Bose symmetry, and thus, is less “super”. “Super” things are cool, and thus, we prefer SUSY instead that boring Fermi-Bose symmetry naming! The final enlightenment, through some king of French connections thanks to Scherk, allows us to discover SUSY and SUGRA (part of superstring/M-theory models) as the answer to the existence of fermionic fields in Nature. If you feel disappointing at this stage you should give up and not to read any more. However, mathematics is the best tool we have to find the Nature deeper secrets. Are you ready for them? I hope so…


Making a transition phase is not easy. If your path is seriousness, let us consider a small and incomplete set of definitions (dictionary) of the word super as many people ask: “Why do you call it supersymmetry/SUSY?” If you also read my previous list http://www.thespectrumofriemannium.com/2013/05/08/log102-superstuff-the-list/ you will find other alternative and complementary superword set. Many “superwords” are scientific, some others being of common knowledge origin. For instance:

“Super”: word of american (?) origin. Opposite of “regular”. In Switzerland, “super” is a small additional invest compared to “regular” (~1.08 SF/liter vs. ~1.04 SF/liter).

“Scientific superwords”: supernova, superstar, superhelix (DNA), superconductivity, superfluidity, superaerodynamics, superphosphate, superpolyamide, supersonic, superstructure, superoxide, superactinide, superbrane, super p-brane, supermembrane, supercontinent, supermaterial, superspace, superstring, superparticle, superheterodyne, superhuman, superatom, supermolecule, supergroup, superalgebra, superextendon, supermatrix, supersymmetry, supergravity,superfield, supervielbein, super-Higgs mechanism,…

“Political superwords”: superpowers (in 1979 language: Monaco, Lietchtenstein), superphoenix (mythological bird unknown to the Antiquity, of Gallic origins).

“Touristic superwords”: superTignes (where is the CERN staff today?).

“American colloquial superwords”: “Gee, it’s super”, “Super-Duper”.

“Pop music superwords”: supertramp.

“Daily life superwords”: supermarket (Migros, Coop), superman (comic stripo by F. Nietzsche).

“Poetic touch”: superlove; the French poet Jules SUPERvielle (Montevideo 1884-Paris 1960) wrote in 1925 a collection of poems called “Gravitations”.

SUSY and SUGRA, and their relatives/offsprings, are just a few modern superwords added to this list. SUSY in flat spacetime is described by some elements:

  • Supersymmetric transformation laws (SSTL/S.S.T.L.):
    a set of continuous transformations changing classical, commuting, Bose real (or complex)  integer spin fields into classical anticommuting (sometimes dubbed classical anticommuting  c-numbers), Grassmann Fermi variables with half-integer spin fields and vice versa. Roughly, you can think about this as

    (1)   \begin{equation*} \delta \vert\mbox{Boson}\rangle\sim \vert\mbox{Fermion}\rangle,\;\;\; \delta \vert\mbox{Fermion}\rangle\sim \vert\mbox{Boson}\rangle\end{equation*}

  • SUSY model/theory.  A classical lagrangian/action field theory, in flat spacetime, whose action is invariant under SUSY transformations. Generally speaking, you hear about N=1, N=2, N=4 or even N=8 theories often in the superworld, but you can also build theories with any arbitrary number of supersymmetry generators. Mathematically speaking, your freedom is your (consistent, free of contradictions) imagination!
  • Extended SUSY. Popular expression for SUSY theories with N>1.
  • SUSY (a.k.a. supersymmetry). Any field of M\Lambda \Phi which studies local and rigid supersymmetric transformation laws (SSTL), supersymmetric theories/models and extended SUSY. Theoretical physicists are crazy for (finding) SUSY and adding SUSY to their theories. It is likely an obsession. Experimentally, SUSY is hard to kill, but LHC and other future colliders, also other cool experiments, are pushing forward into finding SUSY. SUSY is a beast. If SUSY does exist, it has to be broken in Nature. How and where is the puzzle. From the mathematical side, SUSY is known to be the only (almost unique) way to relate bosons and fermions through symmetry, evading the celebrated Coleman-Mandula theorem including fermionic generators in the algebra. That is, SUSY (superPoincaré group to be more precise) is (up to some uncommon exceptions) the only non-trivial extension of the Poincaré group containing the internal symmetries of the Standard Model. That is why some people is resistent and resilient to give up SUSY in these times. Other alternatives, like N-graded Lie algebras, quantum groups are seen just as odd or just as certain special representations of (extended/generalized) SUSY.
  • Superspace. A set of generalized coordinates Z^A, containing both, ordinary spacetime coordinates X^\mu, Bose classically commuting coordinates, and \theta^i_\alpha, Grassmann Fermi classically anticommuting coordinates. Here, \mu=1,2,\ldots,D, i=1,2,\ldots,N.         \alpha can also runs from 1 to D, but there are other alternatives and it is a free index right now. Thus, we get                                                                                     

    (2)   \begin{equation*} \boxed{Z^A=\left(X^\mu,\theta^i_\alpha\right)}\end{equation*}

  • Superfield. Any function \Phi(Z^A) with or without indices.
  • Supermultiplet. Any irreducible representation of SUSY (simple N=1, or extended N>1), expressed in terms of the (super)fields.
  • Matter supermultiplet. Any supermultiplet with J_{max}=1,1/2. You can also generalize this definition to other “matter” (dark?), like J_{max}=3/2 or higher, but it is not usual. J_{max}=1 are called the vector supermultiplets, J_{max}=1/2 are called scalar supermultiplets.
  • Supersymmetric Yang-Mills theories/Super Yang-Mills theories for short (SYM). The self-interacting (g\neq 0) field theory based on a vector supermultiplet, or extended YM theory having SUSY symmetries.
  • Goldstino. The m=0, J=1/2 fermions associated with the spontaneous symmetry breaking of SUSY.
  • Gluinos. The SU(3)_c octets with J=1/2 in SYM theories.
  • Superalgebras. Graded Lie algebras (GLAs), dubbed into a cool superword friendly synonim by crazy M\Lambda \Phi addict physicists. These addict physicists could be named superphysicists…

Example: N=1 SYM theory in D-dimensions. It contains the next ingredients,

1st. Spectrum: A_\mu^i, J=1, real fields; \chi^i, J=1/2 Majorana fields.

2nd. Coupling constant: g\neq 0.

3rd. Infinitesimal SUSY parameter:  \varepsilon. J=1/2 constant Majorana field.

4th. SUSY transformations:

(3)   \begin{equation*} \delta A_\mu^j=i\overline{\varepsilon}\gamma_\mu \chi^j \end{equation*}

(4)   \begin{equation*} \delta \chi^j=\sigma^{\mu\nu}F_{\mu\nu}^j\varepsilon \end{equation*}

5th. Action and lagrangian:

    \[S_{SYM}=\dfrac{1}{g^2}\int d^Dx \mathcal{L}_{SYM}\]

(5)   \begin{equation*} \mathcal{L}_{SYM}=-\dfrac{1}{4}F_{\mu\nu}^jF^{\mu\nu j}+\dfrac{i}{2}\overline{\chi}^j\gamma^\mu \mathcal{D}_{\mu j k}\chi^k \end{equation*}

and where

    \[\boxed{F_{\mu\nu}^j=\partial_\mu A_\nu^j-\partial_\nu A_\mu^j+gf^{j}_{mn}A_\mu^m A_\nu^n}\]

    \[\boxed{\mathcal{D}_{\mu r s}=\partial_\mu \delta_{rs}+g\delta_{rs}^l A_\mu^l}\]

If now, we upgrade SUSY by changing our flat (super)spacetime into curved (super)spacetime, we enter into the SUGRA realm. Firstly, let me introduce the main SUGRA superwords:

  • SUGRA models/theories. Any SUSY model/theory built in curved spacetime invariant under N=1 SUSY (however the latter case is sometimes named extended supergravity, but terminology is just a choice in some cases; nevertheless, pure SUGRA generally refers to N=1).
  • Extended SUGRA models/theories. As suggested in the previous definition, any theory/model built in curved spacetime invariant under N>1 SUSY. The issue of allowing or not curvature in the fermionic sector is open. Curved supertheories are usually more subtle but can be handled, in principle, with suitable tools.
  • SUGRA. The field of M\Lambda \Phi physicists (superphysicists?). It studies local SUSY transformation laws, superfields, supergravity and extended SUGRA theories.
  • Pure SUGRA. SUGRA or extended SUGRA uncoupled to any matter supermultiplets, with a self coupling \kappa, dimensionally M^{-1}.
  • Gravitino. The Rarita-Schwinger field quanta. Spin 3/2 particle associated with local simple (or extended!) SUSY.
  • Super Higgs effect/mechanism. The analogue of the Higgs effect/mechanism for SUSY. Wherever SUSY is spontaneously broken, a gravitino (or several gravitini) eats up a goldstino (or several goldstinos) becoming massive fields.

Example (II): N=1 SUGRA. The ingredients of this theory/model are

1st. Spectrum. Vielbein, J=2 real field V_\mu^a. Gravitino, J=3/2 Majorana field \Psi_\mu^a.

2nd. Coupling constant. \kappa. Related to G_N(D). It has several normalizations.

3rd. SUSY parameter. varepsilon; J=1/2 X-dependent. Majorana field.

4th. SUSY transformations. We have

    \[\boxed{\delta V_\mu^a=-i\kappa \overline{\varepsilon}\gamma^a\Psi_\mu}\]

    \[\boxed{\delta \Psi_\mu=\kappa^{-1}\mathcal{D}_\mu\varepsilon}\]

5th. Action and lagrangian.

(6)   \begin{equation*} S_{SG}=\int d^Dx\mathcal{L}_{SG} \end{equation*}

(7)   \begin{equation*} \mathcal{L}_{SG}=-\dfrac{1}{4\kappa^2}VV^{\mu a}V^{\nu b}\mathcal{R}_{\mu\nu ab}-\dfrac{1}{2}\epsilon^{\mu\nu\rho\sigma}\overline{\Psi}_\mu \gamma_5\gamma_\nu\mathcal{D}_\rho\Psi_\sigma \end{equation*}

and where we define

    \[\boxed{\mathcal{R}_{\mu\nu ab}=\partial_\mu\omega_{\nu ab}+\omega_{\mu a}^c\omega_{\nu cb}-(\mu\leftrightarrow \nu)}\]

    \[\boxed{\mathcal{D}_\nu=\partial_\nu+\dfrac{1}{2}\omega_{\nu ab}\sigma^{ab}}\]

Technical details: there are some technicalities in all these things…SUSY/SUGRA and generally field theory involves different types of spinors. Giving up the weirdest types, there are 3 main types of spinors. These spinors are named Weyl, Dirac and Majorana fields (spinors). The esential features come from the Clifford algebra of spacetime

(8)   \begin{equation*} \{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu} \end{equation*}

where \eta^{\mu\nu}=\mbox{diag}(\underbrace{+,\ldots,+}_{t-times},\underbrace{-,\ldots,-}_{s-times}), D=t+s, and the irreducible representation of the \gamma matices are 2^{\left[D/2\right]} dimensional.  The Majorana representation (MR) is any representation of \gamma matrices in which they are all i times real matrices. A celebrated theorem states that a MR does exist in even D if and only if s-t=0,2\mbox{mod} \;\;8. The charge conguration matrix, C, verifies C\gamma^\mu C^{-1}=-\gamma^{\mu T}, where T means transpose matrix. Majorana spinors are any spinor such that \chi=C\gamma^0\chi^*. If t=1, Majorana spinors exist only if a MR does exist. In the MR, we have C=\gamma^0 and \chi=\chi^*. Weyl spinors are any spinor set for even D, such that \gamma^{D+1}=\eta \gamma^0\gamma^1\cdots\gamma^{D-1}, with \eta such as (\gamma^{0\mu})^2=+1, so Weyl spinors satisfy the conditions \gamma^{D+1}\chi=\pm\chi, and they are “two-component like”. Finally, you can combine these conditions, and build a Majorana-Weyl spinor, i.e., \chi spinors that are both, Majorana and Weyl spinors. They do exist in D even if and only if s-t=0\mbox{mod}\;\;8. Either of these two restrictions will cut the number of independent spinor components in half. And thus, we have seen that in some dimensions it is possible to have both Weyl and Majorana spinors simultaneously. This reduces the number of independent spinor components to a quarter of the original size. Spinors without any such restrictions are called Dirac spinors. Which restrictions are possible in which dimensions comes in a pattern which repeats itself for dimensions D modulo 8. Secretly, this phenomenon is connected to Bott’s periodicity, but I will not discuss it here today. The dimensionality of a Dirac spinor as a solution to the Dirac equation in D spacetime dimensions is given by the dimension of the Dirac matrices. In the familiar example of four dimensions, Dirac spinors belong to reducible representations of the Lorentz group. For arbitrary spacetime dimensions one might wonder what dimensionality an irreducible spinor has got.

Remark: The most general free spinor (from Clifford bilinears up to some exotics) action reads

(9)   \begin{equation*} \boxed{S(\Psi)=\int d^Dx \left(\alpha_1\overline{\Psi}\gamma^\mu\partial_\mu\Psi+\alpha_2m\overline{\Psi}\Psi+\alpha_3m_{5}\overline{\Psi}\gamma^{D+1}\Psi\right)} \end{equation*}


Superalgebras are the heart of SUSY. There are 3 main types of SUSY algebras used in SUSY: superconformal algebras (SCA; 15+N^2 Bose generators, and 8N Fermi generators), super-dS (or super-AdS) algebra (SdSA has 10+N(N-1)/2 Bose generators and 4N Fermi generators); and finally the super-Poincaré algebra (SPA has the same number or generators than the SdSA, except in the presence of the so-called central charges). Some relatives from these superalgebras are:

  • SCA: in flat space, you can get super-\lambda\phi^4 theories, and SYM theories (for N=1,2,4). In curved space, you can form SC SUGRA.
  • SdSA: in curved space (there is no flat space cases here) you can get extended (N>1) or simple N=1 dS SUGRA/AdS SUGRA with SO(N) gauge group.
  • SPA: in flat space, you can obtain virtually all supersymmetric models with N=1, and SYM with D=6,10. In curved space, D=4 SUGRA and extended SUGRA theories with SO(N) gauging. D=11, N=1 is the so-called maximal SUGRA (if you give up higher spin excitations with J>2). The most interesting models are the extreme cases, such as that with N=8 SUGRA in D=4 and N=1 with D=11. The N=4 SYM theory (GSO model) is also very interesting. This model has remarkable properties: it is renormalizable, and the Fermi-Feynman gauge is fully 1-loop finite (no renormalization is needed). The 1-loop corrections to the propagator vanish identically (finite and infinite parts), and finally the Gell-Mann-Low \beta (g) function vanishes identically for 1 and 2 loops. This fact makes the GSO model to exhibit no spontaneous symmetry breaking of SUSY, while coupled to SUGRA it does. Another advantage is that this model compared to the SO(8) theory is that the gauge group is arbitrary and can be fitted for Nature. SCA has additional applications. The superconformal SUGRA, whose bosonic sector contains the Weyl theory of gravity (the tensor R_{\mu\nu}^2-R^2/3 correction is present) and has four derivatives in the action, which leads to dipole ghosts. This is generally used to reject this theory on these principles, well as on the classical ground of one word. The SCA has no dimensional constant entering it and thus, the field theories based on it have dimensionless coupling constants, ensuring renormalizability (SC gravity and SYM). The SdSA contains one dimensional coupling constant m, with dimensions of mass/energy. Translations do not commute and give a rotation times m^2. The Universe described by the SdSA is thus not a flat Universe, but a de Sitter one (SO(3,2) rather than SO(4,1)). The radius of the dS Universe is roughly R_0\sim m^{-1} in natural units. The constant m plays the role of a mass term via the term m\overline\Psi_\mu\sigma^{\mu\nu}\Psi_\nu for the gravitino field, and a cosmological constant m^2/\kappa^2 occurs in the lagrangian, where \kappa is related to G_N, the Newton constant of gravity. In extended theories, with N\geq 2, the vector fields such as A_\mu couple with the dimensionless minimal coupling constant g, and \kappa m\sim g. The SO(3,2) AdS Universe is the simple, maximally supersymmetric solution of the field equations, and the actual size of the Universe introduces a stringent bound on g, such as g\leq 10^{-120}! Actually, there is also another viewpoint of this deduction. We can plug g\sim 1 and g^2/\kappa^4\sim (10^{19})GeV^4 and this fits observations even when we have no idea of why this is so. This picture is related to the spacetime foam by Wheeler, Hawking ad Townsend. As they use to point out, the dS theory has solutions where spacetime is flat or nearly flat at large distances (greater than 1 cm) but very strongly curved at distances of the order of Planck’s length (about 10^{-35}m or similar). The physical spacetime may well be a statistical (emergent!) ensemble of fluctuating spacetime foams (with unknown quantum degrees of freedom) of arbitrary sizes and topologies. The issue of the change of topologies in this fluctuating spacetime is not solved in any current candidate of quantum gravity. And finally, SPA, with or without central charges, have the dimension of a mass, and representations of the SPA with central charges occur for massive supermultiplets only (e.g., N=2, J_{max}=1/2) or for the classical solutions of the N=2,4 SYM theories. In these theories, we have respectively 2 and 6 central charges. In the N=2 model, the 2 charges are the electric and the magnetic charges, satisfying the Montonen-Olive relationship


    Olive remarkable idea was to identify mQ and mG with the momenta P_5, P_6 in certain Minkovskian spacetime (s=5, t=1; D=6) where M_6^2=0. Similarly, the N=4 model suggests to define 6 extradimensional central charges Q_i, and these can be identified with 6 extradimensional momenta with P_i=mQ_i and the mass relationship M_{10}^2=0. These facts are not surprising, since the N=2,4 D=4 theories can be obtained by dimensional reduction from the N=1, D=6,10 SYM theories!


We are going to discuss only the representations of the SPA in D=4 dimensions (s=3, t=1), in the massless case, in terms of fields. One can show that only one of the supercharges Q^i_\alpha is relevant (e.g. Q^i_1), and that it decreases the helicity by 1/2. The particle content of a supermultiplet can be easily found. In the next table, we observe the particle contents of N=1,\ldots,10 supersymmetric free field theories with J_{max}\leq 5/2. Scalar multiplets exist up to N=2, vector multiplets up to N=4, SUGRA multiplets up to N=8 and hypergravity with J_{max}=5/2 multiplets up to N=10.

Table 1. Representation contents of N=1,…10 SUSY with 0 mass. Ogievetsky multiplets (having J_{max}=3/2) are shown up to N=4, but exist up to N=6. Hypergravity multiplets (J_{max}=5/2) are shown to exist from N=1 to N=10 but are shown only for N=9 and N=10.


Explicit constructions are necessary to show that that interacting field theories based on these multiplets do exist. This is tru up to J_{max}=2, but no interacting field theory (hypergravity) based on J_{max}=5/2 exists. Hyper seems to suggest that these theories may not exist, but who knows? Hyperspace was a fantasy before the 19th and 20th century. Recently, hypergravity theories, higher spin theories and hypersuperspace theories have been considered by those M\Lambda \Phi crazy physicists you all know…Remarkably, hyperbole itself comes from certain Greek demagog of Samos, he threatened the comic poets Alcibiade and Nicias of ostracism,but he was turned into ridicule by them an was himself ostracized in 417 B.C.


The super prefix associated with SUSY and SUGRA is due first to superunification of fundamental forces somehow. SUSY and SUGRA own good features and they are specially recognized, in some models, by being renormalizable (cf. classical general relativity, a non-renormalizable theory).

1st. Unification goodness.

-Fields of different spins are unified in the same representation, overcoming previous no-go theorems (e.g., the Coleman-Mandula theorem).

-Internal symmetries and spacetime symmetries are unified. SUSY is (almost) unique but with a lot of models/theories to explore. What is the right model/theory? That is an experimental problem.

-Fermions and bosons play symmetrical roles.

-The dichotomy between fields and sources is also solved.

2nd. Renormalization goodness.

-If the bosonic sector of certain supersymmetric field theory renormalizable, so is the whole theory. Further in that case, the associated supersymmetric field theory is more convergent than its bosonic sector. E.g.: the N=4 SYM theory.

-Non supersymmetric theories of gravity with matter interactions (J=0,1/2;1) and their associated matter fields are one loop non-renormalizable, while pure extended SUGRA theories based on the SPA (N=1,…,8) are 1 and 2 (at least, current knowledge improves these arguments) loop renormalizable and finite. Superconformal gravity is also renormalizable, but so is conformal gravity.


SPA exists for any s,t  signature and any spacetime dimension (D=s+t). If we keep D=4, and increase N, we meet the limits N=2,4,8 for the existence of multiplets with J_{max}=1/2,1,2. Similarly, if we keep N=1, and we increase D, we meet the limits D=6,10,11 for the existence of the same multiplets.

An example of supersymmetric field theory in D=10 (s=9, t=1) is easy to provide. Let us take the example fo the N=1 SYM theory. If we keep N=1 and plug in D=10 in the notations we have introduced here, the theory is still SUSY invariant, provided that the \chi^j are Majorana-Weyl spinors, which is possible since s-t=8. The vector A_M^i in D=10 gives in D=4 a vector A_\mu^i and the MW spinor \chi^i reduces to 4 Majorana spinors \chi^i_j, j=1,2,3,4. This is precisely the field contents of the N=4, D=4 SYM theory! What we have done is dimensional reduction. Dimensional reduction consists in starting from certain higher dimensional theory, usually certain supersymmetric field theory with D\geq 5. One lets the size of the internal dimensions shrink into zero size (radius), in which case only the excitations of the fields which are constant in the extra dimensions are relevant, or have a simple y dependence (of the type \Psi(x,y)\sim \exp imy \Psi (x)) so only certain extra dimensional excitations do survive. The resulting D=4 theory is still supersymmetric (invariant under SUSY), but SUSY may be spontaneously broken. The mass m appears as the momentum in the 5th (or any other) extra dimension/direction. This process can be generalized. In generalized dimensional reduction, a phase \exp (imy) dependence is introduced, as in the usual Kaluza-Klein theory. The N=8 theory can then be generalized starting from the D=5, N=8, J_{max}=2 theory which has a rank 4 group of invariance, namely Sp(8). Going from 5 to 4 dimensions, 4 mass parameters m_i, i=1,2,3,4 can be introduced and a 4-parameter family of N=8 theories is obtained which contain the CJ (Cremmer, Julia) theory as a special case (m_i=0). In these theories, SUSY is spontaneously broken, the Bose-Fermi mass degeneracy is lifted but the breaking is “soft” as the following mass relations still hold

    \[\boxed{\sum_{J=0}^{J=2}\left(-1\right)^{2J}\left(2J+1\right)\left(M_J^2\right)^k=0,\forall k=0,1,2,3}\]

These mass relations imply that the one loop correction to the cosmological constant is still fully finite, but apperently non-zero (note that from this viewpoint, these theories do predict -or posdict- a non null cosmological constant). The spectrum of the N=8 theory is the content of the next section.

7. N=8.

In the spontaneously broken N=8 theory, every massive state is complex and the theory has a U(1) summetry. This extra symmetry is actually a local one. Its gauge filed is the vector field A_\mu^n, obtained by reducing the metric tensor from 5 to 4 dimesnions and 2\kappa A_\mu^\curlywedge=V_\mu^5. In the next section, we shall see that the coupling of this vector field leads to the phenomenon of “antigravity” (but be aware of what it does mean!), and we shall denote the particle associated with the quantized A_\mu^\curlywedge by a curlywedge symbol. (In the original papers, it was used the egyptian symbol “shen”, but it is pretty similar to curly wedge, and it is adorable too!).

If every m_i is equal to the same value, the only symmetry of the theory will be U(1) (apart from SUSY, coordinate invariance,…). If we set all the m_i=m, the spectrum is SU(4)\times U(1) degenerate and at zero mass, there are 15+1 gauge bosons. As a gauge group SU(4) is too small to include the SM group but if we disregard the weak interactions, we obtain a model which includes SU(3)_c\times U(1)\times U(1), that is strong interaction plus electromagnetism plus antigravity. Actually, the breaking of SU(4) into SU(3)_c\times U(1) is automatic if we set m_i=m with m_i=m for i=1,2,3 and m_4=M. The electric charge is taken to be equal to 1/3 for the triplet of J=3/2 graviquarks of mass m, 0 for the J=3/2 singlet gravitino of mass M. Once this is accomplished, all the masses and charges are derived by taking the product of representations. This model contains only one electron of mass 3m, but no muon or tau particles. Even the neutrino fields are absent. this is not too surprising and one could tentatively attribute the family structure to composite bound states as well as the SU(2) of weak interactions is taken into account. A more humble attitude is to take the model just a model not as a final theory. As it stands, it is not too bad: it is 1 and 2 loop finite (unlike Einstein’s theory of gravity coupled to leptons, quarks, and the SM group bosons and particles). It also includes a massless graviton, 8 massless gluons, a photon, an electron, a d type quark (Q=-1/3), a u type quark (Q=+1/3) and a c type quark (Q=2/3). The exotic particles are: the sexy quarks (the sextet 6, with Q=1/3), the gluinos of Fayet (Q=0), a triplet of d-graviquarks (the triplet 3: Q=-1/3), a set of 2 neutral gravitini (1,0), massless scalar gluons (2 octets),  massless singlet scalar particles (2 of them) and massive sextets and singlet scalars, and massive scalar quarks (that we could name “sarks”).


Let us consider the scattering of two particles of mass M_1,M_2 having also a coupling to a massless vector field A_\mu^\curlywedge (this vector field is NOT the electromagnetic potential of the SM, so it is some kind of dark electromagnetism) with charges g_1,g_2. In the static limit the total potential energy is given by

(10)   \begin{equation*} V(r)=4k^2 M_1M_2r^{-1}\left[M_1M_2-k^{-2}g_1g_2\right] \end{equation*}

Scherk’s antigravity is defined by the cancellation which would happen if one had systematically the relation between g’s and M’s

(11)   \begin{equation*} \boxed{g=\epsilon k m} \end{equation*}

and where \epsilon=+1,-1,0 for particles, antiparticles and neutral particles (like Z bosons, neutrinos and others). In 1977, it was guessed that since in N=2 SUGRA there is a vector A_\mu, it had to couple minimally to fields with a gauge coupling constant g\sim km. The coupling was actually written in lowest order in k. Later, in 1978, K. Zachos coupled N=2 SUGRA to the multiplet (1/2(2), 0(4)), with a mass m and found \vert g\vert=km, as well as antigravity (the above cancellation phenomenon!). In 1979, the spontaneously broken N=8 theory was found and it was discovered that a vector A_\mu coupled to all the model fields with strength \vert g\vert=2km, a relation that holds for all the 256 states of the model. If there is such an antigravitational force in Nature, and this is an inescapable consequence of SUSY if N>1 (that is, extended SUSY contains antigravity), why don’t see it? Perhaps, we have already seen and we don’t realize it! If we look at the spontaneously broken N=8 model, we may find a beginning of an answer. Suppose we consider the static force between 2 protons (uud bound states). As the graviton couples no to the mass but to the energy-momentum tensor T_{\mu\nu} it sees the total energy of the quarks and gluons, i.e., the mass of the proton. It is mostly the kinetic energy of the quarks and gluons as a whole. This force contribution is given roughly by the term k^2r^{-2}M_p^2, where M_p is the proton mass. The antigraviton (sometimes called graviphoton) is coupled to \overline{\chi}\gamma^\mu\chi times km, where m is the mechanical mass of the quark is question, and \overline{\chi}\gamma^\mu\chi is the conserved electromagnetic current. Therefore, the antigraviton/graviphoton sees directly the quark mass and not the proton mass and its contribution is \pm k^2r^{-1}(2m_u+m_d)^2. Furthermore, we observer that the \dfrac{\mbox{antigraviton}}{\mbox{graviton}} relative contribution is of the order (m_u/M_p)^2, and that is small, about 10^{-4} for u,d quarks. Finally, if we compute the relative difference between the acceleration of a proton and a neutron, we find that it is given by the expression 3m_u(m_d-m_u)/M_p^2, which is even smaller, depending on the u, d mass difference. In the limit of exact SU(2) symmetry for strong interactions, this difference vanishes so that antigravity is in good health…However, if one looks closer, one finds that the \curlywedge exchange leads to serious problems with the equivalence principle! Let me explain it better. Two atoms of atomic numbers A_1, A_2, having Z_1, Z_2 protons and of net charge zero fall with DIFFERENT accelerations towards the Earth! The force between these 2 atoms is given by

(12)   \begin{equation*} F=8\pi G r^{-2}\left[ M(Z_1,A_1)M(Z_2,A_2)-M^0(Z_1,A_1)M^0(A_2,Z_2)\right] \end{equation*}

The negative term is due to the antigraviton/graviphoton exchange; one generally has:

(13)   \begin{align*} M(Z,A)=Z(M_p+m_e)+(A-Z)M_n\\ M^0(Z,A)=Z(2m_u+m_d+m_e)+(A-Z)(m_u+2m_d)\\ \gamma (Z,A)=8\pi G r^{-2}M(Z_1,A_1)\left[1-\dfrac{M^0(Z_1, A_1)M^0(Z_2,A_2)}{M(Z_1,A_1)M(Z_2,A_2)}\right] \end{align*}

If A_2,Z_2 represent the Earth, we can safely replace \dfrac{M^0}{M}(Z_2,A_2) by 3m_u/M_p. Therefore, the acceleration of (Z_1,A_1) towards the Earth is given by

(14)   \begin{equation*} \boxed{\gamma (A_1,Z_1)=a(A_1,Z_1)=\dfrac{8\pi G}{r^2}M_E\left(1-\dfrac{M^0(A_1,Z_1)}{M(A_1,Z_1)}\dfrac{3m_u}{M_p}\right)} \end{equation*}

The relative difference in acceleration of the 2 atoms will be

(15)   \begin{equation*} \Delta=\dfrac{\delta a}{a}=\dfrac{\gamma (Z_1,A_1)-\gamma (Z_2,A_2)}{\gamma (Z_1,A_1)} \end{equation*}

(16)   \begin{equation*} \Delta=\xi\left[m_e(M_n-m_u-2m_d)+\dfrac{2}{3}(m_u+m_d)(M_n-M_p)+\dfrac{1}{2}(m_u-m_d)(M_n-M_p)\right]\Xi \end{equation*}

and  where \xi=\left( Z_2A_1-Z_1A_2\right)\dfrac{3m_u}{M_p}, \Xi=M^{-1}(Z_1,A_1)M^{-1}(Z_2,A_2). Int the last bracket, the dominant contribution is given by m_eM_n, so \Delta reads

(17)   \begin{equation*} \Delta\sim \left(Z_2A_1-Z_1A_2\right)\dfrac{3m_u}{M_p}\dfrac{m_eM_n}{M_1M_2} \end{equation*}

Putting into numbers this expression, we find that \Delta\sim 10^{-6}, bigger than the most accurate bound on the violation of the equivalence principle. This is unaceptable. Is antigravity wrong after all? Theorists do NOT give up a general idea so easy…In order to save the antigravity idea/phenomenon (something quite general in some BSM theories), one must assume that the antigraviton/graviphoton acquires a mass, likely through the Higgs mechanism. This is rather what it happens, since it (the graviphoton/antigraviton) is universally coupled to scalar fields through the lagrangian

(18)   \begin{equation*} \mathcal{L}=-\dfrac{1}{4}F_{\mu\nu}^\curlywedge F^{\mu\nu\curlywedge} -\dfrac{1}{2}\vert \left( \partial_\mu-ikm A_\mu^\curlywedge \right)\phi\vert^2+m^2\phi^*\phi-V(\phi) \end{equation*}

At the classical level, the v.e.v. (vacuum expectation value) is \langle\phi\rangle\neq 0 and m_\curlywedge=km_\phi\langle\phi\rangle. If \langle \phi\rangle\neq 0 is due to SU(2)xU(1) breaking, one has typically \langle \phi\rangle\sim 1=100GeV, and with \langle \phi\rangle\sim 1GeV one finds that m_\curlywedge\sim 10^{-19}GeV. This gives to the antigraviton a Compton wavelength of the order 1 km, which seems to be reasonable. In the case where m_\curlywedge\neq 0, the potential between an atom with (Z,A) and the Earth is provided by the following expression

(19)   \begin{equation*} \boxed{V(Z,A)=\dfrac{8 \pi G}{R_E}\left( 1-\dfrac{M^0}{M}(Z,A)\dfrac{3m_u}{M_p}\exp\left(-m_\curlywedge R_E\right)\right)} \end{equation*}

This formula would be correct if the Earth were a point like object. Taking into account of its actual size leads (for an homogeneous sphere) to multiplying the last term in this expression by a form factor f(m_\curlywedge R_E), where

    \[f(x)=3x^{-3}\left[x\cosh x-\sinh x\right]\]

The altitude from the surface now appears rather than the distance from the center and it leads to the upper bound on m_\curlywedge^{-1}, given by m_\curlywedge^{-1}\leq 1km. Usually, masses are thought to be fixed parameters. However, one knows that they depend upon external conditions such as the temperature T. If one could “heat the vacuum” enough, the phase where \langle \phi\rangle=0 and m_\curlywedge=0 would be restored. Antigravity devices of this kind however still belongs to the field of Ufology and Sci-Fi (Science-Fiction), and apparently not to the field of mathematical physics.


It will be short and we leave it to a great American hero…

Super manofsteel1Remark: SUSY haters have also their propaganda

2yLoLSUSY2215SUSYhopeless2215but also SUSY has big fans/addicts/superphysicists that counter it…


10. APPENDIX: the simplest SUSY.

If you have read up to here, I am going to give you another “gift”. The simplest SUSY you can find from supermechanics. I will add some additional final questions too ;). Let me begin with certain lagrangian. For simplicity, take the mass of the particle equal to the unit (i.e., plug in m=1). The symplest SUSY lagrangian is then build in when you add to the free particle lagrangian (m=1) the Grassmann part as follows:

(20)   \begin{equation*} \boxed{L_{BF}=L_B+L_F=\dfrac{\dot{x}_\mu\dot{x}_\mu}{2}+\dfrac{i\Psi_\mu\dot{\Psi}_\mu}{2}} \end{equation*}

The Bose part corresponds to translations and the Fermi part correspond to spins. This lagrangia is, in fact, a special case where both, angular momentum and spin angular momentum, are invariant under INDEPENDENT rotations in the variables x,\Psi. Any interacting extension from this free case involves that this lagrangian generalization will be inivariant only under SIMULTANEOUS rotation of x,\Psi. In particular, this lagrangian is invariant under

(21)   \begin{eqnarray*} \delta x_\mu=\omega_{\mu\nu}x_\nu\\ \delta \Psi_\mu=\omega_{\mu\nu}\Psi_\nu \end{eqnarray*}

The L_{BF}=L_{SUSY} has variables with the following Poisson algebra

(22)   \begin{equation*} \{x_\mu,x_\nu\}_{PB}=\{p_\mu,p_\nu\}_{PB}=0 \end{equation*}

(23)   \begin{equation*} \{x_\mu,p_\nu\}=\{\Psi_\mu,\Psi_\nu\}=\delta_{\mu\nu} \end{equation*}

The Hilbert space on which this objects acts is given by \mathcal{H}=L^2(\mathbb{R}^2)\otimes \mathbb{C}^{2^N}, where N=\left[d/2\right]. Thus, under quantization, you obtain that the hamiltonian is certain laplacian operator on \mathbb{R}^d. Generally, up to a sign, you write

(24)   \begin{equation*} H=\dfrac{\hat{p}^2}{2}=\Delta/2 \end{equation*}

Remark: the sign convention is important in some applications. It is generally better, for convergence issues, choose the laplacian so that the eigenvalues are asymptotically positive.

The Noether charge for L_{SUSY} under rotations can be easily work out, and it yields the tensor

    \[Q_{\mu\nu}\equiv J_{\mu\nu}=x_\mu p_\nu-x_\nu p_\mu-\dfrac{i}{2}\left(\Psi_\mu\Psi_\nu-\Psi_\nu\Psi_\mu\right)=L_{\mu\nu}+S_{\mu\nu}\]

This is a good thing. We recover the classical result that rotational invariance implies the conservation of angular momentum J=L+S=angular part+Spin part. In particular, for d=3, we obtain


and it confirms known results of angular momentum under quantization! Now, the full simplest SUSY transformations in action action onto our lagrangian L_{SUSY}=L_{BF}. The most general field-coordinate variation of this lagrangian provides

(25)   \begin{equation*} \boxed{\delta L_{SUSY}=\dot{x}_\mu \delta \dot{x}_\mu+\dfrac{i}{2}\left(\delta \Psi_\mu\right)\dot{\Psi}_\mu+\dfrac{i}{2}\Psi_\mu\left(\delta \dot{\Psi}_\mu\right)} \end{equation*}

Introduce elementary SUSY transformations

(26)   \begin{align*} \delta x_\mu=i\epsilon \Psi_\mu\\ \delta \Psi_\mu=-\epsilon\dot{x}_\mu \end{align*}

Plug in these variations into the \delta L_{SUSY} variation, and operate it to obtain

(27)   \begin{align*} \delta L=i\dot{x}_\mu\epsilon\dot{\Psi}_\mu-\dfrac{i}{2}\epsilon\dot{x}_\mu\Psi_\mu-\dfrac{i}{2}\Psi_\mu\epsilon\dot{x}_\mu=\\ =i\dot{x}_\mu\epsilon\dot{\Psi}_\mu-\dfrac{i}{2}\epsilon\dot{x}_\mu\dot{\Psi}_\mu-\dfrac{i}{2}\dfrac{d}{dt}\left(\Psi_\mu\epsilon\dot{x}_\mu\right)+\dfrac{i}{2}\dot{\Psi}_\mu\epsilon\dot{x}_\mu=\\ =-\dfrac{i}{2}\dfrac{d}{dt}\left(\Psi_\mu\epsilon\dot{x}_\mu\right) \end{align*}

The conserved charge (supercharge is used often) is

(28)   \begin{equation*} \epsilon Q=i\epsilon p_\mu\Psi_\mu=i\epsilon \Psi_\mu p_\mu=i\epsilon \Psi_\mu\dot{x}_\mu \end{equation*}

that is, the Noether supercharge, SUSY generator is

(29)   \begin{equation*} \boxed{Q=i\Psi_\mu\dot{x}_\mu=i\Psi_\mu p_\mu=ip_\mu\Psi_\mu} \end{equation*}

In even dimension, d=2n, we usually quantize the Poisson brackets with the aid of the canonical commutators and anticommutators given by

(30)   \begin{equation*} \left[\hat{x}_\mu,\hat{p}_\nu\right]=i\delta_{\mu\nu} \end{equation*}

for bosons and

(31)   \begin{equation*} \{\hat{\Psi}_\mu,\hat{\Psi}_\nu\}=\delta_{\mu\nu} \end{equation*}

for fermions. Defining

(32)   \begin{equation*} \Psi_\mu=\dfrac{\hat{\gamma}_\mu}{\sqrt{2}} \end{equation*}

it turns that the fermion anticommutator is secretly a (rescaled) Clifford algebra in disguise, since Clifford algebras are defined as

(33)   \begin{equation*} \{\gamma_\mu,\gamma_\nu\}=\{\hat{\Psi}_\mu,\hat{\Psi}_\nu\}=2\delta_{\mu\nu} \end{equation*}

The remaining problem is to find and determine the gamma matrices or some good representation of them. The gamma matrices act onto the Hilbert space \mathcal{H}=L^2(\mathbb{R}^{2d}\otimes\mathbb{C}^{N}, N=2^n in even dimensions, but it can be generalized to odd dimensional spaces too (with care!). Generally speaking, this factorization of the Hilbert space says that SUSY acts on a superpace being “translations times spin”. The quantized Noether operator \hat{Q} associated to SUSY transformations reads

(34)   \begin{equation*} \hat{Q}=i\hat{\Psi}_\mu\hat{p}_\mu=i\dfrac{\gamma_\mu}{\sqrt{2}}\left(-i\partial_\mu\right)=\dfrac{1}{\sqrt{2}}\gamma^\mu\partial_\mu \end{equation*}


(35)   \begin{equation*} \boxed{\hat{Q}=\dfrac{1}{\sqrt{2}}\gamma^\mu\partial_\mu} \end{equation*}

This result teaches us something really cool and amazing: the SUSY quantum mechanical Noether supercharge (operator) is nothing but the Dirac operator (here, acting on the manifold \mathbb{R}^{2n} times the spin group). Remember: the SUSY supercharge is generally speaking certain Dirac-like (Clifford) operator, the product of the Clifford gamma matrix and certain derivative. Indeed, there is something really beautiful in addition to this thing. SUSY transformation can be computed for this operator as well, with new amazing results:

(36)   \begin{align*} \delta Q=i\left(\delta \Psi_\mu\right)\dot{x}_\mu+i\Psi_\mu\delta \dot{x}_\mu=\\ =i(-\epsilon\dot{x}_\mu)\dot{x}_\mu+i\Psi_\mu (i\epsilon\dot{\Psi}_\mu)=-i\epsilon x^\mu\dot{x}^\mu+\epsilon\Psi_\mu\dot{x}_\mu=\\ =-2i\epsilon\left[\dfrac{\dot{x}_\mu\dot{x}_\mu}{2}+\dfrac{i\Psi_\mu\dot{\Psi}_\mu}{2}\right]=-2i\epsilon L_{BF}=-2i\epsilon L_{SUSY} \end{align*}


(37)   \begin{equation*} \boxed{\delta Q=-2i\epsilon L_{SUSY}} \end{equation*}

Motto: the variation of the supercharge is proportional to the SUSY lagrangian (times a constant).

Moreover, compute two successive SUSY transformations, with parameters \epsilon_1,\epsilon_2. Then, you can show that the commutor (and its associated Poisson bracket) reads

(38)   \begin{equation*} \delta_{\epsilon_2}\delta_{\epsilon_1}-\delta_{\epsilon_1}\delta{\epsilon_2}=-2i\epsilon_1\epsilon_2\dfrac{\partial}{\partial t} \end{equation*}

But \partial_t is the generator of translations in time associate to the energy or hamiltonian of the system! This can be easily proved

(39)   \begin{align*} \{\epsilon_2ip_\mu\Psi_\mu,\epsilon_1ip_\nu\Psi_\nu\}_{PB}=(i)^2p_\mu p_\nu\epsilon_2\epsilon_1\{\Psi_\mu,\Psi_\nu\}_{PB}=\\ -p_\mu p_\nu\epsilon_2\epsilon_1\left(-i\delta_{\mu\nu}\right)=i\epsilon_2\epsilon_1 p^2=-2\epsilon_1\epsilon_2 H \end{align*}

and where we have used that 2H=p^2. Thus, under quantization,

(40)   \begin{align*} \{Q,Q\}=2\hat{Q}^2=2(i\hat{p}_\mu\hat{\Psi}_\mu)(i\hat{p}_\nu\hat{\Psi}_\nu)=\\ -2\hat{p}_\mu \hat{p}_\nu\hat{\Psi}_\mu\hat{\Psi}_\nu-2\hat{p}_\mu \hat{p}_\nu\dfrac{1}{2}\left(\hat{\Psi}_\mu \hat{\Psi}_\nu+\hat{\Psi}_\nu\hat{\Psi}_\mu\right)=\\ -\hat{p}_\mu\hat{p}_\nu \delta_{\mu\nu}=-p^2=-2\hat{H} \end{align*}

Thus, the SUSY supercharge is, generally speaking, “the square root” operator of the hamiltonian, since

(41)   \begin{equation*} \boxed{\{\hat{Q},\hat{Q}\}=-2\hat{H}} \end{equation*}

or equivalently

    \[\boxed{\hat{Q}^2=-\hat{H}} \longleftrightarrow \boxed{\hat{Q}=\sqrt{-\hat{H}}}\]

In summary, the main formulae from the simplest SUSY lagrangian are given by

(42)   \begin{align*} \boxed{L_{BF}=L_B+L_F=\dfrac{\dot{x}_\mu\dot{x}_\mu}{2}+\dfrac{i\Psi_\mu\dot{\Psi}_\mu}{2}}\\ \boxed{\delta L_{SUSY}=\dot{x}_\mu \delta \dot{x}_\mu+\dfrac{i}{2}\left(\delta \Psi_\mu\right)\dot{\Psi}_\mu+\dfrac{i}{2}\Psi_\mu\left(\delta \dot{\Psi}_\mu\right)}\\ \boxed{\delta x_\mu=i\epsilon \Psi_\mu\;\;\;\delta \Psi_\mu=-\epsilon\dot{x}_\mu}\\ \boxed{\hat{Q}=i\hat{\Psi}_\mu\hat{p}_\mu=i\dfrac{\gamma_\mu}{\sqrt{2}}\left(-i\partial_\mu\right)=\dfrac{1}{\sqrt{2}}\gamma^\mu\partial_\mu}\\ \boxed{\delta Q=-2i\epsilon L_{SUSY}}\\ \boxed{\delta L=\dfrac{\epsilon}{2}\dfrac{dQ}{dt}}\\ \boxed{\{\hat{Q},\hat{Q}\}=-2H}\\ \boxed{\hat{Q}^2=-\hat{H}} \longleftrightarrow \boxed{\hat{Q}=\sqrt{-\hat{H}}} \end{align*}

Finally, some exercises for addict, eagers readers…I do know you do exist and you are OUT there!

1) Generalize this discussion to a simple manifold with a metric. SUSY covariance reads from

    \[\Psi_\mu=\dfrac{\partial x^\mu}{\partial x'^\mu}\Psi '^\nu\]

and the metric is

    \[ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu\]

The invariant object

    \[ Q=i\langle \dot{x},\Psi\rangle=ig_{\mu\nu}\dot{x}^\mu \Psi^\nu\]

is well defined. Note that it depends on the velocities \dot{x}^\lambda. The lagrangian of this exercise is provided by a covariant version of our simple L_{SUSY}:



    \[L=\dfrac{\langle \dot{x},\dot{x}\rangle}{2}+\dfrac{i}{2}\Bigg\langle\Psi,\dfrac{D\Psi}{Dt}\Bigg\rangle\]

Here, we define the Christoffel connection

    \[\Gamma^\nu_{\lambda\mu}=\dfrac{1}{2}g^{\nu\rho}\left(\partial_\lambda g_{\rho\lambda}+\partial_\mu g_{\lambda\rho}-\partial g_{\lambda\nu}\right)\]

The covariant derivative is given by

    \[\dfrac{D}{Dt}=\dfrac{d}{dt}+\dot{x}^\lambda \Gamma_\lambda\]

and from the expansion

    \[\Gamma^\mu_\nu=dx^\lambda \Gamma^\mu _{\lambda \nu}\]

we can get

    \[ \boxed{R^{\mu}_{\;\;\;\nu}=d\Gamma^{\mu}_{\;\;\; \nu}+\Gamma^\mu_{\;\;\;\sigma}\wedge \Gamma^\sigma_{\;\;\;\nu}}\]

and curvature components

    \[ R^{\mu}_{\;\;\;\nu}=\dfrac{1}{2}R^{\mu}_{\;\;\;\nu\rho\sigma} dx^\rho\wedge dx^\sigma\]

2) Find the Euler-Lagrangian equations for this covariant generalization of our simple lagrangian, L_{SUSY}. Is there any conservation law there? Reason your answer.

3) Express the Dirac operator for any general curved manifold M in local coordinates.

Have fun!!!!!!!

 Let SUSY, SUGRA and ANTIGRAVITY be with you!!!!!!!

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