Relaxing Symmetries in Field Theory:

from Noether theorem to noncommutativity and

the challenges to Lorentz invariance

Alfredo Iorio

Graduate Lecture Notes

IPNP Prague

25 October - 29 November, 2005

Contents

Introduction 4

1 Spacetime and Internal Symmetries 61.1 Emmy Noether’s Theorem . . . . . . . . . . . . . . . . . . . . 61.2 The issue of “improvements” . . . . . . . . . . . . . . . . . . 8

1.2.1 Maxwell’s −14F 2. The canonical and symmetric en-

ergy momentum tensor . . . . . . . . . . . . . . . . . 81.2.2 The scale vs conformal symmetry case . . . . . . . . . 8

1.3 How to deal with broken symmetry . . . . . . . . . . . . . . . 91.4 The quantum case . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Combining Spacetime and Internal Symmetries 132.1 No-go Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 O’Raifeartaigh Theorem(s) . . . . . . . . . . . . . . . . . . . 152.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 The (special) case of SUSY 183.1 The Haag-Lopuszansky-Sohnius Theorem . . . . . . . . . . . 18

3.1.1 Building up the Super-Poincare Algebra. . . . . . . . . 183.1.2 Irreducible Representations and Supermultiplets. . . . 21

3.2 General features of SUSY theories . . . . . . . . . . . . . . . 243.3 The Wess-Zumino model and its (Noether) supercharges . . . 273.4 N=2 Susy Classical Georgi-Glashow model and the Mass For-

mula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 The infinite number of degrees of freedom of quantum fields,the Haag’s theorem and the issue of locality 34

2

5 Spatiotemporal Noncommutativity 355.1 The fundamental length . . . . . . . . . . . . . . . . . . . . . 355.2 Examples in Nature . . . . . . . . . . . . . . . . . . . . . . . 355.3 Formalization: The gauge-covariant approach and the Seiberg-

Witten map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.4 Problems? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 From Lorentz symmetry violation to Lorentz symmetry en-hancement? 36

7 The elusive effects of Lorentz violation: the case of −14 F 2 37

3

Introduction

These are days of a deep rethinking of fundamental issues in theoreticalphysics, the most important of all being the symmetry principles. In theselectures we want to introduce the challenges to Lorentz invariance that arisefrom different areas of the theoretical investigation – especially the hypoth-esis of noncommuting coordinates – and lead to everyday more precise ex-perimental tests. This is a particularly fortunate case, as lately the mostadvanced theoretical research in particle physics has been decoupled fromthe experimental challenges.We shall try to accomplish our goal by first presenting the most importanttools and ideas used in the application to physics of the symmetry principle.In this way we shall have the chance to state and prove Emmy Noether’s1918 theorem in field theory, discuss to the greatest possible extent thesubtleties involved with space-time symmetries, look closely at the differencebetween internal and space-time symmetry, and introduce what ought tobe the highest possible symmetry of the S-matrix, namely Supersymmetry(SUSY). This will close the first part of the course, where the old materialabove described will be introduced in the most (to us) critical way.Once we have solidly established our own technology on the fundamentalissues we shall be able to move to the new arena of what is known these daysas “Lorentz violation”. I shall first try to explain the theoretical motivationsbehind the investigation of a departure from Lorentz invariance. Then I shalltouch upon as many as possible theoretical approaches to this fascinatinghypothesis. I will then focus on one of them, namely noncommutativity ofthe spatiotemporal coordinates, for two reasons: a) it is what I understandand like the most; b) there are very recent important results I want todisclose to you. Eventually I will keep my promise of telling you about thestate-of-the-art with experiments and experimental proposals.The lectures are two hours each. There are two (easy) exercises for eachlecture. I will propose a series of papers for further reading on the varioustopics we will discuss, the most important papers will be listed separately.

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The expected home-work is to solve the exercises and present a proposedpaper in the form of a journal club during the last class.

Further reading

Of course, it is purely a matter of taste to pick up one or two references onthe symmetry principle in physics! On the other hand, I do have my ownsuggestions:

[1] K. Brading and E. Castellani Eds., “Symmetries in Physics: Philo-sophical Reflections”, 2003, Cambridge University Press. This wonderfulbook is a collection of novel contributions to the philosophical debate onsymmetry in physics. Do not get misled by the word “philosophy”, theyknow a lot about mathematics and physics...

[2] L. O’Raifeartaigh, “The Dawning of Gauge Theory”, 1997, PrincetonUniversity Press. This is a rare book on the history of physics wrote bya first rank physicist. The early papers on gauge theory are reprinted andcommented upon in a deep scholarly way. Again, it is not just a book onthe history of the gauge principle, there is a lot to learn from the physicsside!

5

Chapter 1

Spacetime and InternalSymmetries

1.1 Emmy Noether’s Theorem

In a theory with Lagrangian density L(Φi, ∂Φi) (where the collective indexi takes care of the different fields, as well as their spin type) the Noethercurrent for space-time transformations has the form

Jµf = ΠµiδfΦi + Lδfxµ , (1.1)

where δfxµ = fµ, Πµi = δL/δ∂µΦi, δfΦi = Φ′i(x)−Φi(x), and the infinites-imal quantities fµ take the following form

fµ = aµ or fµ = ωµν xν or fµ = axµ or fµ = aµx2−2a·xxµ , (1.2)

for infinitesimal translations, rotations (and boosts), dilations, and specialconformal transformations, respectively, where, as usual, ωµν = −ωνµ.The expression for the current (1.1) has been obtained by varying the action,including the measure, under an arbitrary space-time transformation, andonly afterwards one tests the invariance. Let us sketch here a proof.Let us consider the Action

AΩ =∫

Ωd4xL(Φi, ∂Φi) (1.3)

where Ω is the space-time volume of integration. The infinitesimal transfor-mations of the coordinates, of the fields and of the derivatives of the fieldsare given respectively by

xµ → x′µ = xµ + δfxµ (1.4)

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Φi(x) → Φ′i(x′) = Φi(x) + δ∗fΦi(x) (1.5)

∂µΦi(x) → ∂′µΦ′i(x′) = ∂µΦi(x) + δ∗f∂µΦi(x) (1.6)

note that δ∗f does not commute with the space-time derivatives.When we act with this transformation the Action changes to

A′Ω′ =∫

Ω′d4x′L(Φ′i, ∂

′Φ′i) . (1.7)

If the transformation is a symmetry we have A′Ω′ − AΩ = 0, for any fieldconfiguration, i.e. off-shell, therefore at the first order we obtain

0 = A′Ω′ −AΩ

=∫

Ωd4x

[(1 + ∂ρδfxρ)

(L(Φi, ∂Φi) +

∂L∂Φi

δ∗fΦi +∂L

∂(∂µΦi)δ∗f∂µΦi

)− L

]

=∫

Ωd4x

( ∂L∂Φi

δ∗fΦi +∂L

∂(∂µΦi)δ∗f∂µΦi + L∂ρδfxρ

)(1.8)

where (1 + ∂ρδfxρ) is the Jacobian of the change of coordinates from x′ tox at the first order.Let us now introduce another variation δf that commutes with the deriva-tives1:

δfΦi = Φ′i(x)− Φi(x) = LfΦi , (1.9)

where Lf is the standard Lie derivative along the vector field fµ.If we do so we can write

δ∗fΦi = ∂µΦi(x)δfxµ + δfΦi and δ∗f (∂µΦi) = ∂µ∂νΦi(x)δfxν + δf∂µΦi .(1.10)

Substituting these back in (1.8) we obtain∫

Ωd4x

[∂µ

( ∂L∂(∂µΦi)

δfΦi

)+

( ∂L∂Φi

∂µΦi +∂L

∂(∂νΦi)∂µ∂νΦi

)δfxµ + L∂µδfxµ

]

=∫

Ωd4x

[∂µ

( ∂L∂(∂µΦi)

δfΦi

)+

( ∂L∂xµ

δfxµ + L∂µδfxµ)]

(1.11)

=∫

Ωd4x(E.L.)iδfΦi ,

which finally gives the wanted conservation law on-shell ∂µJµ = 0 where

Jµ = Πµi δfΦi + Lδfxµ

1Sometimes in literature, these transformations are referred to as “geometrical” trans-formations.

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Although this current is all we need to write our (conserved) quantities, tomake more transparent the appearance of known quantities we can writeback the space-time dependent variations δ∗fΦi we obtain

Jµ = Πµi δ∗fΦi − (Πµ

i ∂νΦi − ηµνL)δfxν

= Πµi δ∗fΦi − Tµνδfxν (1.12)

that leads, for fµ = aµ, to the definition of the canonical (i.e. in generalnon-symmetric, non-traceless, etc.) energy-momentum tensor Tµν , and, forfµ = ωµνxν , to the definition of the three rank tensor of angular momentumMµνλ.Noether theorem can also be stated in a different perspective, namely bychecking that δL = ∂µV µ, which is only true for invariant actions, and thenwriting the current as

Jµ = Nµ − V µ (1.13)

where, Nµ ≡ Πµiδ∗fΦi is the rigid current, and we recall that in this caseδ∗fΦi = Φ′i(x′)− Φi(x).The choice δ∗f is the most useful in the case of gauge transformations andSupersymmetry. In both cases it is not possible to write V µ = −Lδfxµ: forSUSY that is because there is no δfxµ; for gauge symmetry this is clearlydue to the fact that they are internal symmetries, hence do not involvespace-time transformations, and δfxµ = 0.

1.2 The issue of “improvements”

1.2.1 Maxwell’s −14F 2. The canonical and symmetric energy

momentum tensor

See handwritten notes.

1.2.2 The scale vs conformal symmetry case

The virial current for the action A(Φ, ∂Φ), with Φ any field

Jµ = ΠνΛµνΦ (1.14)

where Πµ = δA/δ∂µΦ, Λµν = dΦgµν + 2Σµν , and Σµν is the appropriatespin-connection.The general conditions for conformal invariance of a scale-invariant theoryare

for d ≥ 3 Jµ = ∂νJµν for d = 2 Jµ = ∂µJ (1.15)

8

It is interesting to see what happens if we allow the kinetic term of theLiouville field to be multiplied by scalar fields. Consider for example

A =∫

d2x

(12ηab

[h(φ)∂aθ∂bθ + hαβ(φ)∂aφ

α∂bφβ]− eθV (φ)

)(1.16)

where the φ-fields are conformal scalars and eθ has scale dimension deθ =−2. In this case the virial current is Jµ = h(φ)∂µθ and thus is not a totalderivative. It follows that although the action is rigid scale invariant it isnot conformally invariant, and if it is Weyl gauged the Weyl gauging cannotbe replaced by Ricci gauging. Thus, even for actions which are quadraticin the derivatives of scalar fields, rigid scale invariance does not necessarilyimply conformal invariance.

1.3 How to deal with broken symmetry

With the current (1.1) one can: i) test whether the given transformation isa symmetry by picking the correspondent fµ in (1.2), and checking whether∂µJµ = 0, by using the equations of motion; ii) use the Noether charges Qf ≡∫

d3xJ0f , and the canonical equal-time Poisson brackets Φi(x), Πj(y) =

δji δ(~x − ~y), to generate the transformations of an arbitrary function of the

canonical variables

G(Φi, Πi), Qf ≡ ∆fG(Φi, Πi) . (1.17)

Note also that, for f0 = g0 = 0 ,

Qf , Qg = Q[f,g] , (1.18)

and Eq.s (1.17) and (1.18) hold whether or not ∂0Qf = 0. Of course, whenQf acts on the fields it must reproduce the transformations one started with∆fΦi = δfΦi.Take the action

I = −14

∫d4x [FµνFµν − 1

2θαβFαβFµνFµν + 2θαβFαµFβνF

µν ] . (1.19)

We will learn more on this action later in the course. For the moment allwe need to know is that θαβ = −θβα and real.The Noether currents for space-time transformations are

Jµf = ΠµνδfAν + Lfµ , (1.20)

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where the fµs are given in (1.2), Πµν = δL/δ∂µAν , and, being Πµαβ =

δL/δ∂µθαβ = 0, the transformations δfθαβ do not enter the Noether current.Let us now analyze the symmetry properties by writing the divergence ofthis current as

∂µJµf = ΠµνFαν∂µfα − L∂µfµ . (1.21)

We obtain∂µJµ

f = 0 translations (1.22)

for infinitesimal translations fµ = aµ, and

∂µJµf = ωα

µΠµνFαν homogeneous Lorentz (1.23)

for infinitesimal rotations and boosts,

∂µJµf = a(ΠµνFµν − 4L) dilations (1.24)

for infinitesimal dilations

∂µJµf = 2ΠµνFαν(aαxµ−aµxα−a ·xδα

µ)+8a ·xL special conformal (1.25)

infinitesimal special conformal transformations. For θµν = 0, Πµν = −Fµν ,and from (1.22)- (1.25) one immediately sees that ∂µJµ

f = 0 for all fµsin (1.2), and the theory is invariant under the full conformal algebra (asexpected for classical electromagnetism!).On the other hand, the action I, for θµν 6= 0, is only invariant under transla-tions. Of course, this leaves room to special choices of the parameters and/orof the field configurations, to obtain conserved currents. For instance, if oneperforms two dependent infinitesimal boosts with parameters ω0

1 = ω02 = ω

, from the condition in (1.23) one obtains Π12(E2 − E1) = E3(Π13 + Π23).Thus, for instance, one finds conservation if the electric field ~E lives in the(1, 2)-plane, and has equal components.We now want to check the dynamical consistency of the transformations,along the lines of what explained earlier. It is straightforward to see thatfor Aµ and Fµν

∆fAµ = δfAµ and ∆fFµν = δfFµν , (1.26)

while for θµν

∆fθµν = 0 , (1.27)

for all fµs in (1.2).

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At this point one has to investigate whether θµν could be treated as a La-grange multiplier (as e.g. in supersymmetric models). The answer in no. Ifone uses

δI

δθµν= 0 , (1.28)

as a constraint this implies Fµν = 0, which is too trivial a theory (puregauge).Note that if one uses (1.28) the expressions of ∂µJµ

f in (1.22)-(1.25)are triv-ially zero for all the space-time transformations.We conclude that δfθµν cannot be fixed by any symmetry requirement (withthe only exception of translations), and that to have a physical meaningfultheory one should not make use of the “equations of motion” (1.28). There-fore, among all possible δfθµνs that represent the conformal algebra, themost natural choice is

δfθµν = ∆fθµν = 0

(which agrees with the translation invariance), and θµν does not transformunder dynamically consistent space-time transformations.

1.4 The quantum case

See handwritten notes.

1.5 Exercises

Exercise I.a Derive the angular momentum tensor Mµνρ for the Maxwelltheory −1

4F 2 and compare its energy momentum tensor with the one ob-tained from translational symmetry.Exercise I.b Prove that the virial current is a pure divergence for theMaxwell theory (thus obtaining that scale invariance implies full conformalinvariance in this case).Exercise I.c Suppose that the only nonzero components of θµν is θ3 (whereθ0i = 0 and θij = εijkθk with i, j, k = 1, 2, 3). In this case, which subgroupof the Lorentz group SO(3, 1) survives in the theory analyzed in Section1.3?Exercise I.d Discuss the meaning of the quantum relation [H, Q] = 0, fora theory with Hamiltonian H and a charge Q. Can you give an explicitexample?

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1.6 Further Reading

E. Noether, Invarianten beliebiger Differentialausdruke, Nachr. d. Konig.Gesellsch. d. Wiss. zu Gottingen, Math-phys. Klasse (1918) 37;J. Lopuszanski, “An introduction to Symmetry and Supersymmetry in quan-tum field theory”, World Scientific, 1991;A. Iorio, T. Sykora, On the Space-Time Symmetries of NoncommutativeGauge Theories, Int.J. Mod. Phys. A 17 (2002) 2369;A. Iorio, L. O’Raifeartaigh, I. Sachs and C. Wiesendanger Weyl-Gaugingand Conformal Invariance, Nucl. Phys. B 495 (1997) 433;P. G. Federbush, K. A. Johnson, Phys. Rev. 120 (1960) 1926.

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Chapter 2

Combining Spacetime andInternal Symmetries

2.1 No-go Theorems

The ”no-go theorems” prove the impossibility of non-trivially combiningLorentz invariance and internal symmetry for physical theories. The mostpowerful is the Coleman-Mandula (CM) theorem [1]. The generalization ofthis theorem lead to the discovery of supersymmetry.The statement of the CM theorem is as follows: “Let E be a connectedsymmetry group of the S matrix, and let the following five conditions hold:(I) E contains a subgroup locally isomorphic to the inhomogeneous Lorentz(Poincare) group L; (II) all particle types correspond to positive-energy rep-resentations of L, and, for any finite mass M , there are only a finite numberof particle types with mass less than M ; (III) elastic-scattering amplitudesare analytic functions of the center of mass energy and of the momentumtransfer in some neighborhood of the physical region; (IV) at almost all ener-gies, any two plane waves scatter; (V) the generators of E are representableas integral operators in momentum space, with kernels that are distributions.Then E is locally isomorphic to L × T , the direct product of the Poincaregroup and the internal symmetry group”.In the 1960’s there were two kinds of motivations for investigating this prob-lem in particle physics.The first concerned the mass splitting occurring within the multiplets ofparticles. The challenge was to find a group containing the internal symme-try group, with non-trivial commutations among the generators of the latterand those of space-time translations, Pµ. For example let us consider the

13

isospin group, whose SU(2) algebra we write in the Cartan-Weyl basis

[T+, T−] = 2T0 and [T0, T±] = ±T± , (2.1)

where T+ (T−)is the step-up (step-down) operator, and T0 is the generatorof the Cartan sub-algebra. The commutator [Pµ, T+] is zero for ordinaryisospin symmetry. However, if it is different from zero within the biggergroup, this would give account for the mass splitting as a higher symmetryeffect. Consider the doublet of nucleons |n〉 ≡ |+〉 and |p〉 ≡ |−〉 as the twostates of the fundamental irrep of SU(2)

T0|±〉 = ±|±〉 , T±|±〉 = 0 , T∓|±〉 = |∓〉 . (2.2)

By our hypothesis [Pµ, T+] = 0 ⇒ P 2|±〉 = m2|±〉, which means m2p =

m2n. Hence the experimental results mp ∼ 938.3MeV and mn ∼ 939.6MeV,

cannot be explained. Suppose, instead, that

[Pµ, T+] = cµT+ (2.3)

where cµ is orthogonal to Pµ (cµPµ = 0) and commutes with the othergenerators

P 2|+〉 = P 2T+|−〉 = T+

(P 2|−〉+ 2cµPµ|−〉+ c2|−〉

)= T+m2

n|−〉 (2.4)

orP 2|−〉 = (m2

n − c2)|−〉 . (2.5)

i.e. the mass splittingm2

p = m2n −∆m2 , (2.6)

where ∆m2 ≡ c2. Nowadays the accepted explanation of the mass-splittingphenomenon is the breakdown of flavor symmetry.The second motivation to investigate non trivial combinations of space-timeand internal symmetries was the discovery of a model where the internal 3-flavor symmetry group SU(3) and the non-relativistic spin group SU(2) werenon-trivially combined into SU(6), which contains but is not isomorphic toSU(3)×SU(2). This gives the so-called static quark model. As a matter offact the barion octet JP = 1

2

+

n pΣ− Σ0/Λ Σ+

Ξ− Ξ0

14

and decuplet JP = 32

+

∆− ∆0 ∆+ ∆++

Σ∗− Σ∗0 Σ∗+

Ξ∗− Ξ0

Ω−,

which differ by spin, both fit into a 56-plet of SU(6). This is easily seen ifone considers that the dimension of the representation has to be d× (2J +1), where d is the dimension of the SU(3) representation, and J the non-relativistic spin. In this case: i) d = 10 and 2J + 1 = 4 gives 40, while ii)d = 8 and 2J + 1 = 2 gives 16, which add up to 56. On the other hand thetensorial representation of SU(6): 6× 6× 6 = 56 + 70 + 70 + 20. Similarconsiderations hold for the meson nonets of JP = 0− and JP = 1−. Thenatural task then was to extend this result to a fully relativistic theory.These programs were brought to a negative end first by the O’Raifeartaigh(LOR) theorem [2] which completes and generalizes the results of the workstarted with the first “no-go” theorem of McGlinn (McG) [3], and laterby the CM theorem stated above. We shall later prove and discuss LORtheorem in some detail. All these theorems hold if one considers Lie groupsas symmetry groups of the theory (LOR theorem holds for finite order Liealgebras, while CM theorem holds also for the infinite case) and are oflocal nature. Nevertheless if one relaxes the assumption of having onlystandard Lie groups, for instance by allowing for graded structures, then thenegative-type conclusions no longer hold [4]. The most surprising feature ofthese graded algebras is the occurrence of transformations among particlesdiffering by spin: this is the birth of supersymmetry. In Ref. [5] the mostgeneral supersymmetric algebra of the S matrix was introduced and itsrepresentations extensively studied, closing the era of the “no-go” theoremswith a “let’s go” theorem: the Haag-Lopuszanski-Sohnius (HLS) theorem.(Note that the title of the paper in Ref. [5] is “All Possible Generators ofSupersymmetry of the S Matrix” as opposed to the title of the paper in Ref.[1] “All Possible Symmetries of the S Matrix”). In the following we shallstate McG theorem, state and discuss LOR theorem, and finally prove anddiscuss CM theorem.

2.2 O’Raifeartaigh Theorem(s)

LOR theorem. “Let L be the Lie algebra of the inhomogeneous Lorentzgroup, consisting of the homogeneous part M and the translation part P .Let E be any Lie algebra of finite order, with radical R and Levi factor G. If

15

L is a subalgebra of E, then only the following four cases occur: (i) R = P ;(ii) R Abelian but larger than, and containing P ; (iii) R solvable but notAbelian, and containing P ; (iv) R ∩ P = 0. In all cases, M ∩R = 0”. Themain algebraic tools used in this theorem are the Levi decomposition and thefreedom of redefining the generators (the redefinitions have trivial physicalconsequences). Levi’s radical-splitting theorem in Lie algebra theory statesthat any Lie algebra E of finite order is the semidirect sum of a semisimplealgebra G (the Levi factor) and the radical R (an invariant solvable algebra,where solvable means that for some integer k the k-derived algebra is zero).LOR theorem enables one to classify the ways in which L can be embeddedin E. Case (i) is the only physical case and, up to a redefinition, reducesto E = L ⊕ T , where T is a semisimple algebra (internal symmetry). Case(ii) cannot be reduced to the previous direct sum but is unphysical sinceit introduces a translation algebra of more than four dimensions. Case(iii) is the most unphysical since, for non-trivial representations, hermitianconjugation cannot be defined. Case (iv) amounts, up to a redefinition, toembedding L as a subalgebra in a simple Lie algebra. It is again unphysicaldue to the fact that the parameters corresponding to the Pµ have a non-compact range and this lead to serious difficulties in defining multiplets,even in the absence of mass-splitting. Thus, while it is possible to embedL in a larger algebra E, the ways in which this may be done are restrictedand only the direct sum has a clear physical meaning. The McG resultcan be obtained as a special case of LOR theorem by using the first McGassumption alone and the redefinition freedom.If one now considers the Hilbert space H on which any irreducible represen-tation of the group generated by E operates, and if the mass operator P 2

has a discrete eigenvalue m2 and is self-adjoint on H, then the eigenspaceHm belonging to m2 is closed and is invariant with respect to the elementsrepresenting E on H. Hence the elements representing E cannot producethe mass-splitting. Sometimes in literature this result (the mass-splittingtheorem) is referred to as the LOR theorem.

2.3 Exercises

Exercise II.a Show that the product of 3 fundamental (vector) represen-tations of SU(6) boils down to the sum of 4 tensorial (symmetric, antisym-metric and mixed) irreps given in the text 6 × 6 × 6 = 56sym + 70mix +70mix + 20asym. (Hint: use the Young tableaux’ rules for SU(6).)Exercise II.b Explain why for non-relativistic (Galilean invariant) theoriesthe O’Raifeartaigh theorem does not hold. (Hint. This has to do with the

16

compactness of the space-time group.)

2.4 Further Reading

[1] S. Coleman and J. Mandula Phys. Rev. 159 (1967) 1251;[2] L. O’Raifeartaigh Phys. Rev. 139 (1965) 1052;[3] W. D. McGlinn Phys. Rev. Lett. 12 (1964) 467;[4] Yu. A. Gol’fand and E. P. Likhtman JETP Lett. 13 (1971) 323;[5] R. Haag, J. Lopuszanski and M. Sohnius Nucl. Phys. B 88 (1975) 257;[6] S. Weinberg, The Quantum Theory of Fields, Vol. III Supersymmetry,(Chp. 24 ≡ first Chp.)

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Chapter 3

The (special) case of SUSY

3.1 The Haag-Lopuszansky-Sohnius Theorem

3.1.1 Building up the Super-Poincare Algebra.

Let us relax one of the conditions of the CM theorem: introduce anticom-muting generators as possible symmetries of the S matrix. This correspondsto consider a graded algebra with odd generators

Q,Q′, Q′′

besides the ordinary even ones

X, X ′, X ′′ ,

with the algebra among them being given by

[Q,Q′]+ = X [X, X ′]− = X ′′ [Q,X ′]− = Q′′

where the commutator (anti-commutator) is denoted by [, ]− ([, ]+). It isimportant to realize that the CM theorem will not be proved false, it willrather be generalized. Hence within the super-group we will indeed be ableto mix non-trivially internal and space-time symmetries, but in a more subtleway than [Pµ, T+] 6= 0, and the CM results can still be used. For instancethe even generators have to be

L = M −DP

X ∈ or (3.1)A = A1 ⊕A2 ,

18

where L is the Poincare algebra, semidirect product of Lorentz and transla-tions, while the elements of the algebra of internal symmetry A are Lorentzscalars, A1 is semi-simple, and A2 is Abelian.From the point of view of Lorentz properties

Pµ transforms under the (12 , 1

2) vector irrep.

Mµν transforms under the (1, 0) + (0, 1) second-rank tensor irrep.

the elements of A transform under the (0, 0) scalar irrep.

The odd generators instead

Q =∑

Qα1...αa,α1...αb(3.2)

transform under the spin 12(a + b) irrep of Lorentz, and, of course (a + b)

is odd. Suppose that Q belongs to the algebra too, and consider the spin(a + b) object

Q1...1︸︷︷︸a times

, 1...1︸︷︷︸b times

Q 1...1︸︷︷︸a times

,1...1︸︷︷︸b times

If this product has to belong to the algebra hence (a + b) can only be either0 or 1, because

[Q, Q]+ = X

and the spin (a+b) X can either be a scalar (in A) or a vector under Lorentz(the second-rank tensor is ruled out by the hypothesis that Q is odd). Hencethe first anti-commutation relation of this algebra is easily written as

[QLα, QαM ]+ = 2σµ

ααPµδLM , (3.3)

where L, M = 1, . . . , N , and for N > 1 we say that Susy is extended. Notethe space-time nature of the supercharges.In order to construct the other relations one (cleverly) uses the followingsuper Jacobi identities

[A, [B, C]±]± ± [B, [C, A]±]± ± [C, [A,B]±]± = 0 , (3.4)

where the use of the commutator [, ]− or the anti-commutator [, ]+ dependson the even/odd nature of the generators involved, and the signs among theterms depends on the number of commutations of the odd generators.Let us look at the relation

[ QLα︸︷︷︸

( 12,0)

, QMβ︸︷︷︸

( 12,0)

]+ = εαβ Z [LM ]︸ ︷︷ ︸(0,0)

+M(αβ)Y(LM)

︸ ︷︷ ︸(1,0)

(3.5)

19

where we indicate the properties under Lorentz. Since there are no spin 32

generators we have[ QL

α︸︷︷︸( 12,0)

, Pµ︸︷︷︸( 12, 12)

]− = 0 (3.6)

and, by using the super Jacobi identities

0 = [Pµ, [QLα, QM

β ]+]− = [Pµ,Mρσ]−σρσY (LM) ⇒ Y (LM) = 0

since [Pµ,Mρσ]− 6= 0. We can now write the N-extended Susy algebra

[QLα, QαM ]+ = 2σµ

ααPµδLM (3.7)

[QLα, QM

β ]+ = εαβZ [LM ] (3.8)[QαL, QβM ]+ = εαβZ∗[LM ] (3.9)

[QLα, Pµ]− = [QαL, Pµ]− = 0 (3.10)

[QLα, Bl]− = (Sl)L

MQMα (3.11)

[Bl, QαL]− = (S∗l)ML QαM (3.12)

[Bl, Bm]− = iCklmBk (3.13)

where the Bls are the internal symmetry generators belonging to A1, andthe Poincare ”sector” of the algebra has been omitted. Note that, if onewants to consider the largest (super) symmetry of the S-matrix, the fullconformal group has to be included1.By using the super-Jacobi for Bl, Q

Lα, QM

β , it is easy to prove that the Z [LM ]sclose and invariant sub-algebra of A = A1 + A2. By using the super-Jacobi for QL

α, QMβ , QγK , one easily proves that [QγK , Z [LM ]]− = 0 and

12εβα[[QK

α , QNβ ]+, Z [LM ]]− = [Z [KN ], Z [LM ]]− = 0, hence, since from the hy-

pothesis A1 is semi-simple, the Z [LM ]s span A2, the Abelian sub-algebra ofA, [Bl, Z

[LM ]]− = 0.The generators Z [LM ] are central charges of the Susy algebra. They play amajor role in Susy gauge theories, where the Higgs mechanism to give masscan actually take place within Susy, if the central charges are non-zero (seenext Section). It is important to note that, even when at the algebraic levelit is possible to have Z [LM ] 6= 0, as in the N=2 case, is the actual physicalrealization of the algebra (i.e. the underlying model) that tells us if thecentral charges are different from zero. For instance, in the phase where the

1This means that besides the generators of the Poincare group, Pµ and Mµν , also thedilation generator D, and the special conformal generators Cµ have to be considered. Seealso the Chapter on Noether currents.

20

gauge symmetry is not broken, even the N=2 SYM possesses trivial centralcharges.Finally, in the Exercises of this Chapter we propose to prove that the (Sl)L

M sform a representation of A.

3.1.2 Irreducible Representations and Supermultiplets.

We have now a fully relativistic (even conformal) way to combine space-timeand internal symmetries. We already saw that in the static non-relativisticquark model the SU(6) (broken) symmetry combined together particles dif-fering by spin (for instance the spin 1/2 octet and the spin 3/2 decuplet ofbaryons). We have now a more powerful way of doing that within the Susymultiplets.Firstly we notice that for any finite dimensional representation of Susy (sothat the trace is well defined), we can prove that there are an equal numberof bosonic and fermionic degrees of freedom. Note here that we do not needto have an equal number of bosonic and fermionic particles, but only ofdegrees of freedom. To prove that it is just matter of defining a fermionnumber operator NF , so that

(−)NF |BOSE >= +1|BOSE > and (−)NF |FERMI >= −1|FERMI >

We can now compute the following trace

Tr(−)NF [QLα, QαM ]+ = Tr(−)NF QL

αQαM + (−)NF QαMQLα

= Tr−QLα(−)NF QαM + QL

α(−)NF QαM = 0

where the minus sign in the first term of the second line is due to the factthat Qα transforms bosons ↔ fermions, and the second term in the sameline is due to the cyclicity of the trace. By using the first relations of theSusy algebra we then have

2σµααδL

MTr(−)NF Pµ = 0 ⇒ Tr(−)NF = 0 (3.14)

which proves our statement,

(−)NF =

+1. . .

+1−1

. . .−1

. (3.15)

21

The irreducible representations (irreps) of extended Susy are easily found interms of suitable linear combinations of the supercharges QL

α, L = 1, ..., N ,to obtain creation and annihilation operators acting on a Clifford vacuumΩ0. There are three possible cases: I) M 6= 0 and Z = 0; II) M = 0 andZ = 0; III) M 6= 0 and Z 6= 0.

I). Massive, Central-Charge-lessWe move in the rest frame Pµ = (−M, 0, 0, 0), and write the Susy algebraas

[QLα , (QM

β )†]+ = 2M δLMδβ

α (3.16)

[QLα , QM

β ]+ = [(QLα)† , (QM

β )†]+ = 0 (3.17)

where L,M = 1, N . By defining aLα ≡ (2M)−1/2QL

α, and (aLα)† ≡ (2M)−1/2QαL,

we have 2 (for α) × N (for L) = 2 N fermionic annihilation/creation opera-tors, in terms of which the Susy algebra above becomes

[aLα, a†Mβ ]+ = δβ

αδLM [aL

α, aMβ ]+ = [a†Lα , a†Mβ ]+ = 0 . (3.18)

The generic state is then given by

Ω(n)α1

A1· · · αn

An=

1√n!

(aα1A1

)† · · · (aαnAn

)†Ω0 , (3.19)

where aAαΩ0 = 0. Of course, the multiplicity (degeneracy) of this state is(

2Nn

). Thus the dimension of this irrep is

dI =2N∑

n=0

(2Nn

)= 22N . (3.20)

II). Mass-less, Central-Charge-lessTo find the dimension of this irrep it suffices to notice that Pµ = (−E, 0, 0, E)),hence

[QLα, QαM ]+ = 2

(2E 00 0

)

αα

δLM , (3.21)

and zero the others. This means that only one element on the right handside is different from zero, while in the massive case there are two entries ofδβα different from zero. Thus the number of annihilation/creation operators

is reduced to one half with respect to the massive case, and we can concludethat the dimension of this irrep is

dII =N∑

n=0

(Nn

)= 2N , (3.22)

22

or dII =√

dI . The discrepancy of the dimension of the massive and mass-less irreps causes problems in the Higgs mechanism. To cure it one has toconsider the next case.

III). Massive, Central-ChargeIn the rest frame we can write

[QLα , (QM

β )†]+ = 2M δLMδβ

α (3.23)

[QLα , QM

β ]+ = εαβ Z [LM ] (3.24)

[(QLα)† , (QM

β )†]+ = εαβ Z∗[LM ] (3.25)

where L,M = 1, ..., N . Let us consider N even, and take L ≡ (a, l), M ≡(b,m), where a, b = 1, 2, and l, m = 1, ..., N/2.By performing a unitary transformation on the QL

α we can introduce newcharges QL

α = ULMQM

α so that in this basis Z [LM ] is mapped to εab2|Zn|,where Zn = |Zn|eiζn , |Zn| ≥ 0, ζn = 0, n = 1, ..., N/2.We can now define the following annihilation operators

alα =

1√2(Q1l

α + εαγ(Q2lγ )†) (3.26)

blα =

1√2(Q1l

α − εαγ(Q2lγ )†) (3.27)

and their conjugates a†α and b†α, in terms of which we can write the algebraas

[alα , am

β ]+ = [blα , bm

β ]+ = [alα , bm

β ]+ = 0 (3.28)

[alα , am†

β ]+ = δlmδαβ2(M + |Zn|) (3.29)

[blα , bm†

β ]+ = δlmδαβ2(M − |Zn|) (3.30)

For α = β the anticommutators (3.29) and (3.30) are never less than zeroon any states. Therefore from (3.29) follows M + |Zn| ≥ 0 and from (3.30)follows M − |Zn| ≥ 0. By multiplying these two inequalities together weobtain

M ≥ |Zn| (3.31)

The states for which the inequality is saturated are called Bogomolnyi-Prasad-Sommerfield (BPS) states, and we have the so-called short Susymultiplet. The shortness is easily seen by considering a certain numberr ≤ n of BPS states, for which M = |Zi|, i = 1, ..., r. From Eq. (3.30) oneimmediately sees that 2r operators bl

α must vanish. Thus, if r = n = N/2all the N/2 operators bl

α must vanish. This reduces the number of creation

23

and annihilation operators of the Clifford algebra from 2N to N . Thereforethe dimension of the massive representation reduces to the dimension of themassless one: from 22N to 2N , and we can implement the Higgs mechanismwithout breaking Susy: dBPS

III = dII .

3.2 General features of SUSY theories

First let us discuss an example were Susy appears in the most simple way:Susy Quantum Mechanics. These results are due to Witten. For a niceaccount see Ref. [1].

Consider the motion in the direction x of an electron in a magnetic fielddirected along z but function only of x: Bz(x). The hamiltonian

H =1

2m(~p− qe

~A)2 + iqe

2m~∇ · ~A +

|qe|2m

~σ · ~B , (3.32)

where qe and m are the charge and mass of the electron, respectively, andthe natural units are used h = c = 1, with the choice

Ax = 0 = Az Ay =√

m

|qe| W (x) (3.33)

becomes

H =12

(p2

m+ W 2(x) + σ3

1√m

dW

dx

). (3.34)

Now define

Q1 ≡ 1√2(σ1

p√m

+ σ2W ) Q2 ≡ 1√2(σ2

p√m− σ1W ) , (3.35)

and discover that Susy is on its way

[Qi, Qj ]+ = 2δijH [H, Qi] = 0 i, j = 1, 2 . (3.36)

What we have done is to consider ”the square root” of the hamiltonian, in thesame spirit of how Dirac considered the ”square root” of the Klein-Gordonequation (2 + µ2)φ = 0 to obtain his (spinorial) equation (i 6 ∂ + µ)ψ = 0.There are two important considerations to be done here: i) this toy modelSusy shares many of the properties of Susy in field theory; ii) here Susyis implemented in a different spirit respect to the fundamental approach ofconsidering it as a symmetry of the S matrix. Here we list other areas ofphysics where Susy is (or could be) implemented in this spirit:

24

* Nuclear Physics (Iachello)

* Chaotic Systems (Efetov)

* Superconductivity (Nambu)

On a different footing are the following areas, where the Susy scenario hasnot been experimentally seen

0 Standard Model

0 Gravity (Supergravity)

After this brief overview we are ready to discuss the general features ofSusy models, based on the algebraic structure of the symmetry itself. Thehamiltonian of any Susy theory is given in terms of the Susy charges:

4H = σ0αα[Qα, Qα]+ = (Qσ0Q + Qσ0Q) ≥ 0 (3.37)

on any physical state, where the N = 1 case is considered. In particular onthe vacuum |0〉

< 0|H|0 >= 0 i.e. H =: H : . (3.38)

This can be seen by thinking of Susy as a space time symmetry implementedby the group element of the super-Poincare group

G(x, θ, θ) = expi(−xµPµ + θQ + θQ) . (3.39)

In gauge theory without SSB the invariance of a model under a given gaugegroup G is implemented on the action (by construction) and on the vacuum

G|0〉 = eαaT a |0〉 = |0〉hence, T a|0〉 = 0, where T a, with a = 1, ...,dimG, are the generators of thegroup. Similarly in a Susy theory

Qα|0〉 = 0 Qα|0〉 = 0 , (3.40)

which proves our statement about the automatic normal ordering of thehamiltonian.This is a first signal of the nice behavior of Susy theory under renormaliza-tion. As a matter of fact, the normal ordering for a quantum field consistsin the subtraction of the infinite vacuum energy. For a one-dimensionalharmonic oscillator (H.O.)

H|n〉 = (n +12)hω|n〉 → H|0〉 =

hω

2|0〉

25

this subtraction is finite, and allowed. In (free) field theory the vacuumenergy one is discarding with the normal ordering is infinite

Evaccum =∑

k

hωk

2→∞ .

Already Pauli in the 1950s realized that in a theory with an equal number

of bosons and fermions with equal masses ωbosek =

√m2 + ~k2 = ωfermi

k ≡ ωk

the vacuum energy would have been automatically zero, just consider thata fermionic oscillator has negative vacuum energy

HF =∑

k

hωfermik (f †kfk − 1/2) (3.41)

and combine it with the bosonic partner (always there in Susy theories)

HB =∑

k

hωbosek (b†kbk + 1/2) (3.42)

to obtainH = HB + HF =

∑

k

hωk(b†kbk + f †kfk) =: H : (3.43)

The example above given holds for free fields. In fact, the important pointhere is that this phenomenon holds no matter how complicated is the inter-action, even for effective theories, as long as Susy is a symmetry. This isparticularly important if one considers that for interacting field theories thevacuum-to-vacuum (or v2v) graphs < 0|0 > are, in principle, very nastilydivergent. I say in principle because in standard perturbative approachesto QFT it is assumed that < 0|0 >= 1. We will see that this is a strongrequirement, not really satisfied by interacting quantum fields (Haag’s theo-rem), especially in the relativistic regime (as was noticed by Dirac who putit, more or less, this way “The relativistic interaction is to strong to keep theincoming particles within the same Hilbert space as the outgoing particles.This is sort of working for feeble interactions, while for fully relativistic onesthe state representing the incoming particle is kicked out of the Hilbert spaceto a different one” [see the Introduction of Dirac’s lectures on quantum fieldtheories].It is not surprising then that Susy theories have nice renormalization prop-erties. These properties are exploited in full details in the so-called ”non-renormalization theorems” mostly due to Seiberg, even if a better way ofcalling these theorems would be ”no-need-to-renormalize theorems”. Theidea is again based on the fermi-bose symmetry, which one wants to im-plement also at the level of the Feynman graphs. For instance, in Susy

26

QED, the vertex qeψγµAµψ has as Susy counterpart qesψγµAµsψ, wheresψ represents the super-partner of the electron ψ within the supermultiplet,hence has the same mass, but bosonic statistics. The fermionic loop is thencancelled by the bosonic one (same coupling qe and mass).For instance the perturbative contributions to Quantum Super Yang-Mills(SYM) theories are

N=1 Tree Level + 1-Loop + ... (well behaved)

N=2 Tree Level + 1-Loop

N=4 Tree Level

hence there are no quantum corrections at all to the N=4 classical SYM!Note that the instanton, non-perturbative, contributions have not been con-sidered.Note also that in bosonic string theory (the first string theory after Veneziano’shadronic string) the number of space-time (target space) dimensions had todepart from d = 4 to d = 26 to remove a quantum anomaly. This cancelationtakes place, instead, at d = 10 when Susy is implemented.Let us conclude this overview with same nomenclature of the super-partnersof the particles within the Susy Standard Model and Gravity:

• fermions: electron, quarks, ... → sfermions: selectron, squarks, ...

• gauge bosons → gauginos: photino, gluino, ...

• graviton → gravitino

the exception is the neutrino which has as super-partner the neutralino.

3.3 The Wess-Zumino model and its (Noether) su-percharges

The Lagrangian density and supersymmetric transformations of the fieldsfor the Wess-Zumino model are given by

L = − i

2ψ 6∂ψ − i

2ψ 6 ∂ψ − ∂µφ∂µφ† + FF † + mφF + mφ†F † − m

2ψ2 − m

2ψ2

(3.44)and

δφ =√

2εψ δφ† =√

2εψ (3.45)δψα = i

√2(σµε)α∂µφ +

√2εαF δψα = i

√2(σµε)α∂µφ† +

√2εαF †(3.46)

δF = i√

2ε 6 ∂ψ δF † = i√

2ε 6∂ψ (3.47)

27

where φ is a complex scalar field, ψ is its Susy fermionic partner in Weylnotation and F is the complex bosonic dummy field.As explained earlier, in this case one has to use the expression

Jµ = Nµ − V µ

to compute the Noether supercurrents. A note on partial integration in thefermionic sector of (3.44) is in order. We see that ψ and ψ play the doublerole of fields and momenta at the same time. It is just a matter of taste tochoose Dirac brackets for this second class constrained system or to partiallyintegrate to fix a proper phase-space and implement the canonical Poissonbrackets.If one chooses the canonical Poisson brackets, then it is only a matter ofconvenience when to partially integrate the fermions. In fact, even if twoparts of the current Nµ and V µ both change under partial integration, thetotal current Jµ is formally invariant, namely its expression in terms of fieldsand their derivatives is invariant but the interpretation in terms of momentaand variations of the fields is different. Of course both choices give the sameresults, therefore one could either start by fixing the proper phase spacesince the beginning or just do it at the end.Let us keep (3.44) as it stands, define the following non canonical momenta

πµψα =

i

2(σµψ)α πµα

ψ= i

2(σµψ)α (3.48)

πµφ = −∂µφ† πµ

φ† = −∂µφ (3.49)

and use Jµ = Nµ − V µ to obtain the current.We compute V µ by varying (3.44) off-shell, under the given transformations,obtaining

V µ = δφπµφ + δφ†πµ

φ† − δφψπµψ − δφ†ψπµ

ψ+ δF ψπµ

ψ + δF †ψπµψ

−2δFonψπµψ − 2δF †onψπµ

ψ(3.50)

where δXY stands for the part of the variation of Y which contains X (forinstance δF ψ stands for

√2εF ) and Fon, F †

on are the dummy fields givenby their expressions on-shell (F = −mφ†, F † = −mφ). Note here that wesucceeded in finding an expression for V µ in terms of πµ’s and variations ofthe fields. Note also that the terms involving Fon and F †

on were obtainedwithout any request but they simply came out like that.Then we write the rigid current

Nµ = δφπµφ + δφ†πµ

φ† + δψπµψ + δψπµ

ψ(3.51)

28

and the full current is given by

Jµ = Nµ − V µ = 2(δonψπµψ + δonψπµ

ψ) (3.52)

therefore Jµ = 2(Nµ)onfermi, with obvious notation. In the bosonic sector

Nµ completely cancels out against the correspondent part of V µ. In thefermionic sector δF ψπµ

ψ in Nµ cancels out against the term coming fromV µ, δφψπµ

ψ in Nµ and in V µ add up and combined with the 2δFonψπµ

ψ in V µ

gives 2δonψπµψ in the full current Jµ. Similarly for ψ.

Therefore we conclude that:a the dummy fields are on-shell automatically2;

and, if we keep the fermionic non canonical momenta given in (3.48),b the full current is given by twice the fermionic rigid current (Nµ)on

fermi.While the first result is general, the result b is only valid for simple

Lagrangians and it breaks down for less trivial cases. It is interesting to seefor which class of theories it holds.

Now we rewrite Jµ in terms of fields and their derivatives

Jµ =√

2(ψσµσν ε∂νφ + iεσµψFon + h.c.) (3.53)

then choose one partial integration

Jµ = δonψπµIψ +

√2ψσµσνε∂νφ

† + i√

2εσµψF †on (3.54)

or =√

2ψσµσν ε∂νφ + i√

2εσµψFon + δonψπµIIψ (3.55)

where πµIψ = iσµψ (πµI

ψ= 0) and πµII

ψ= iσµψ (πµII

ψ = 0) are the canonicalmomenta obtained by (3.44) conveniently integrated by parts, and performour computations using canonical Poisson brackets.Choosing the setting I, for instance, what is left is to check that the charge

Q ≡∫

d3xJ0(x) =∫

d3x(δonψπI

ψ +√

2ψσ0σνε∂νφ†+ i

√2εσ0ψF †

on

)(3.56)

correctly generates the transformations. This is a trivial task in this casesince the current and the expression of the dummy fields on-shell are verysimple and the transformations can be read off immediately from the charge(3.56). It is worthwhile to notice at this point that to generate the transfor-mations of the scalar field φ† one has to use

δφψπµIψ = δφ†πφ† +

√2ψσ0σiε∂iφ (3.57)

2As a matter of fact using Noether currents we are extracting dynamical informationout of a symmetry realized algebraically via the dummy fields, hence the procedure itselfforces supersymmetry to be realized at a dynamical level putting those non-dynamicalfields on shell.

29

Notice also that the transformation of ψ is obtained by acting with thecharge on the conjugate momentum of ψ: Q , πI

ψ−.

3.4 N=2 Susy Classical Georgi-Glashow model andthe Mass Formula

Edward Witten and David Olive (Phys. Lett. B 78 (1978) 97) consideredthe classical N=2 supersymmetric Georgi-Glashow action with gauge groupO(3) spontaneously broken down to U(1) and its supercharges3.The action for this SU(2) N=2 Super-Yang-Mills (SYM) theory is

A =∫

d4x− 1g2

[14F a

µνFaµν +DµφaDµφa† + iψa 6Dψa + iλa 6Dλa ,

+1√2(ψa[λ, φ]a + h.c.)− 1

2([φ, φ†]a[φ, φ†]a)]− ϑ

64π2F a

µνF∗aµν .

where a = 1, 2, 3 is the gauge group index, SU(2) in our case; for each a, φis the scalar field, ψ, λ are two Weyl fermionic fields, Aµ is a vector field.

The Susy transformations of these fields arefirst Susy, parameter ε1

δ1~φ =

√2ε1 ~ψ

δ1~ψα =

√2εα

1~E (3.58)

δ1~E = 0

δ1~E† = i

√2ε1 6D~ψ + 2i[~φ†, ε1~λ]

δ1~ψα = −i

√2εα

1 6Dαα~φ† (3.59)

δ1~φ† = 0

δ1~λα = −εβ

1 (σµν αβ

~Fµν − iδαβ

~D)

δ1~Aµ = iε1σ

µ~λ δ1~D = −ε1 6D~λ (3.60)

δ1~λα = 0

second Susy, parameter ε2

3We shall consider the gauge group SU(2), since all the local considerations are thesame.

30

δ2~φ = −

√2ε2~λ

δ2~λα = −

√2εα

2~E† (3.61)

δ2~E† = 0

δ2~E = −i

√2ε2 6D~λ + 2i[~φ†, ε2 ~ψ]

δ2~λα = i

√2εα

2 6Dαα~φ† (3.62)

δ2~φ† = 0

δ2~ψα = −εβ

2 (σµν αβ

~Fµν + iδαβ

~D)

δ2~Aµ = iε2σ

µ~ψ δ2~D = ε2 6D~ψ (3.63)

δ2~ψα = 0

We used dummy fields E, D explicitly in the transformations, and thegeneric SU(2) vector is defined as ~X = 1

2σaXa with a = 1, 2, 3, we followthe summation convention, and the covariant derivative and the vector fieldstrength are given by

DµXa = ∂µXa + εabcAbµXc (3.64)

andF a

µν = ∂µAaν − ∂νA

aµ + εabcAb

µAcν (3.65)

Note that the last term is the action above is the instanton term. Thisis a pure divergence of the form ∂µKµ, where

Kµ ∼ εµνρσ(Aaν∂ρA

aσ + εabcAa

νAbρA

cσ)

and when integrated in the action gives the Pontryagin index. Note also,that due to the pseudo-tensor (axial) nature of the dual F ∗

µν , the time-reversal symmetry is broken. That is why the ϑ-angle is related to CPviolation. On this point see, for instance, Ryder’s last chapter, and Wein-berg’s first (discrete symmetries C, P, T) and second (topological objects inQFT, in particular the QCD ϑ-vacua) volumes.The central charge for this theory was obtained by Witten and Olive

Z = i√

2∫

d2~Σ · (~Πaφa +14π

~BaφaD) a = 1, 2, 3 (3.66)

31

where d2~Σ is the measure on the sphere at spatial infinity S2∞, the φa’sare the scalar fields, the ~Ba’s are the magnetic fields, ~Πa is the conjugatemomentum of the vector field ~Aa and φa

D ≡ τφa, with

τ =ϑ

2π+ i

4π

g2, (3.67)

the complex coupling constant whose real and imaginary part are relatedto the CP violating ϑ-angle of the ϑ-vacua, and to the SU(2) coupling,respectively.

In the classical case φaD is merely a formal quantity with no precise

physical meaning. On the contrary, in the low-energy sector of the quantumtheory, it becomes the e.m. dual of the scalar field.In the unbroken phase Z = 0, but, as well known, in the broken phasethis theory admits ’t Hooft-Polyakov monopole solutions. In this phase thescalar fields (and the vector potentials) tend to their vacuum value φa ∼ a ra

r

(Aai ∼ εiab rb

r2 , Aa0 = 0), where a ∈ C, as r → ∞. This behavior gives riseto a magnetic charge. By performing a SU(2) gauge transformation on thisradially symmetric (“hedgehog”) solution we can align < 0|φa|0 > along onedirection (the Coulomb branch), say < 0|φa|0 >= δa3a, and the ’t Hooft-Polyakov monopole becomes a U(1) Dirac-type monopole.In this spirit we can define the electric and magnetic charges as

qe ≡ 1a

∫d2~Σ · ~Π3φ3 (3.68)

qm ≡ 1a

∫d2~Σ · 1

4π~B3φ3 =

1aD

∫d2~Σ · 1

4π~B3φ3

D (3.69)

where aD = τa and only the U(1) fields remaining massless after SSB appear.These quantities are quantized, since4 qm ∈ π1(U(1)) ∼ π2(SU(2)/U(1)) ∼Z and qe is quantized due to Dirac quantization of the electric charge inpresence of a magnetic charge.Thus, after SSB, the central charge becomes

Z = i√

2(nea + nmaD) (3.70)

The mass spectrum of the theory is then given by

M =√

2|nea + nmaD| (3.71)4We say that the ’t Hooft-Polyakov magnetic charge is the winding number of the map

SU(2)/U(1) ∼ S2 → S2∞, that identifies the homotopy class of the map. By considering

the maps U(1) ∼ S1 → S1∞, where S1

∞ is the equator of S2∞, it is clear that a similar com-

ment holds for the U(1) Dirac-type magnetic charge. It turns out that the two homotopygroups, π2(S

2) and π1(S1), are isomorphic to Z.

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We shall call this formula the Montonen-Olive mass formula. It is nowcrucial to notice that this formula holds for the whole spectrum consistingof elementary particles, two W bosons and two fermions, and topologicalexcitations, monopoles and dyons. For instance the mass of the W bosonsand the two fermions can be obtained by setting ne = ±1 and nm = 0,which gives mW = mfermi =

√2|a|, whereas the mass of a monopole (ne = 0

and nm = ±1) amounts to mmon. =√

2|aD|. This establishes a democracybetween particles and topological excitations that becomes more clear whene.m. duality is implemented.

3.5 Exercises

Exercise III.b Show that

[Sm, Sl]KL = iCkml(Sk)K

L (3.72)

i.e. that the Sls form a representation of the internal symmetry algebra ofthe Bls: [Bm, Bl] = iCk

mlBk.Exercise III.a Using the supercharge given in the text, obtain the Wess-Zumino transformations for all the fields, including the dummy fields.

33

Chapter 4

The infinite number ofdegrees of freedom ofquantum fields, the Haag’stheorem and the issue oflocality

See notes.

34

Chapter 5

SpatiotemporalNoncommutativity

5.1 The fundamental length

The issue of locality: the Doplicher et al argument for a fundamental lengthfrom the Heisenberg uncertainties in general relativity and space-time non-commutatitivy.

See Notes

5.2 Examples in Nature

We closely follow the lecture by R.Jackiw, Physical instances of noncom-muting coordinates, hep-th/0110057

5.3 Formalization: The gauge-covariant approachand the Seiberg-Witten map

We closely follow the paper by J.Madore, S.Schraml, P.Schupp, J.Wess,Gauge theory on Nocommutative Spaces, hep-th/0001203

On the Seiberg-Witten map for the Abelian case see notes.

5.4 Problems?

The problem of the infra-red/ultra-violet (IR/UV) connection ? The prob-lem of unitarity with θoi 6= 0?

35

Chapter 6

From Lorentz symmetryviolation to Lorentzsymmetry enhancement?

See notes.References:M. Chaichian, P. P. Kulish, K. Nishijima and A. Tureanu, “On a Lorentz-

invariant interpretation of noncommutative space-time and its implicationson noncommutative QFT,” Phys. Lett. B 604, 98 (2004);

M. Chaichian, P. Presnajder and A. Tureanu, “New concept of rela-tivistic invariance in NC space-time: Twisted Poincare symmetry and itsimplications,” Phys. Rev. Lett. 94, 151602 (2005);

M. Dimitrijevic and J. Wess, “Deformed bialgebra of diffeomorphisms”,arXiv:hep-th/0411224;

P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp andJ. Wess, “A gravity theory on noncommutative spaces”, Class. Quant. Grav.22, 3511 (2005).

36

Chapter 7

The elusive effects of Lorentzviolation: the case of −1

4F2

See notes.References:Z. Guralnik, R. Jackiw, S. Y. Pi and A. P. Polychronakos, “Testing non-

commutative QED, constructing non-commutative MHD”, Phys. Lett. B517, 450 (2001);

P. Castorina, A. Iorio, D. Zappala, Noncommutative Synchrotron, Phys.Rev. D 69 (2004) 065008; On the Vacuum Cerenkov Radiation in Non-commutative Electrodynamics and the Elusive Effects of Lorentz Violation,Europhys. Lett. 71 (2005) 912; Violation of Lorentz Invariance and Dy-namical Effects in High Energy Gamma Rays, Nucl. Phys. B (Proc. Suppl.)136 (2004) 333.

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