ARTICLE

Canonical analysis of a system with fermionic gauge symmetryD.G.C. McKeon

Abstract: A non-abelian gauge field with a topological action is coupled to a spin-3/2 Majorana spinor. The symmetries of thismodel are analyzed using the Dirac constraint formalism. These symmetries include a fermionic symmetry and the algebra ofthese symmetries closes; it is not the algebra of supergravity. The action is invariant without the need to introduce auxiliaryfields.

PACS No.: 11.30.Pb.

Résumé : Un champ de jauge non abélien avec action topologique est couplé a un spineur de Majorana. Nous analysons lessymétries de ce modèle a l'aide du formalisme de contraintes de Dirac. Ces symétries incluent une symétrie fermionique et leuralgèbre est fermée, mais n'est pas l'algèbre de la supergravité. L'action est invariante, sans avoir besoin d'introduire des champsauxiliaires. [Traduit par la Rédaction]

1. IntroductionThe idea of extending the Poincaré group through use of anti-

commuting (fermionic) generators led to both globally supersym-metric and locally supersymmetric (supergravity) theories [1].These models were devised with the purpose of having them beinvariant under a supersymmetry that had been defined at theoutset. However, there is also the possibility of a model havingboth bosonic and fermionic symmetries whose algebra is not thatof supergravity. Such a model has been considered in ref. 2.

In applying the Henneaux–Teitelboim–Zanelli (HTZ) formalism[3] to supergravity in (2 + 1) dimensions [4] to find the symmetries(both bosonic and fermionic) that are present in the model, itbecomes apparent that the same approach can be applied to findsimilar symmetries occurring in a (3 + 1)-dimensional model inwhich a gauge field with a topological action couples to amasslessMajorana spin-3/2 field.

In the next section we will introduce such a model and thenanalyze its symmetries using the HTZ formalism. From the con-straints present in the model, it can be quantized. The conven-tions used appear in Appendix A.

2. The modelThe Einstein–Cartan (EC) action for gravity in (3 + 1) dimensions

is

SEC � � d4xee�me�

nRmn�� (e � det e�

a ) (1)

where e�m is the vierbein and Rmn�� is the field strength associated

with the spin connection wmn�,

Rmn�� � ��wmn� � ��wmn� � wm�k wkn� � wm�

k wkn� (2)

This action can be rewritten as

SEC �1

4� d4x�mnkℓ����em�en�Rkℓ� (3)

In this action, we shall treat e�m and wmn� as independent. (One

could take wmn� to be a function of e�m that is found by solving the

equation of motion for wmn�.)In coupling the gravitational field to matter fields, one employs

the vierbein e�m. However, if one wishes to deal with only the

uncoupled gravitational field, then one could work with the fields

E��mn �

1

4�mnkℓ����ek

�eℓ (4)

so that the EC action takes the topological form

SEC � � d4xEmn��Rmn�� (5)

The EC action in (2 + 1) dimensions when written in terms of thedreibein and spin connection is automatically in topological form

SEC � � d3x ����e�i R��i (6)

where now

R��i � ��w�i � ��w�i � �ijkw�j w�

k (7)

In supergravity, one couples a Majorana spin-3/2 field � to thegravitational field. In (3 + 1) dimensions, this coupling is given by

S3/2 � � d4x���� �e�i �i��� �

i

4w�

mnmn��5 (8)

while in (2 + 1) dimensions we have

Received 31 July 2012. Accepted 11 September 2012.

D.G.C. McKeon. Department of Applied Mathematics, The University of Western Ontario, London, ON N6A 5B7, Canada; Department of Mathematics and ComputerScience, Algoma University, Sault St.Marie, ON P6A 2G4, Canada.

Email for correspondence: dgmckeo2@uwo.ca.

19

Can. J. Phys. 91: 19–22 (2013) dx.doi.org/10.1139/cjp-2012-0324 Published at www.nrcresearchpress.com/cjp on 24 January 2013.

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S3/2 � � d3x �������� �i

2�iw�i�� (9)

In ref. 4, a canonical analysis of supergravity in (2 + 1) dimensions(as defined by (6) and (9)) is performed. Repeating this exercise in(3 + 1) dimensions is considerablymore complicated. (Just definingthe appropriate Dirac bracket (DB) [5] arising from the primarysecond-class constraints that follow from the canonical momentaconjugate to wmni is prohibitively difficult. A canonical analysis ofsupergravity in (3 + 1) dimensions in which a specific gauge ischosen at the outset is given in ref. 7. See also refs. 8 and 9.) Fromthe first-class constraints in (2 + 1)-dimensional supergravity onecan find a set of gauge transformations (both bosonic and fermi-onic) that leave the actions invariant; these transformations havea closed algebra and no auxiliary fields are required. It would be aworthy achievement to obtain the same result for supergravity in(3 + 1) (or indeed, (10 + 1)) dimensions. However, as this enterpriseis so difficult, we will instead consider a model in (3 + 1) dimen-sions in which many of the features of supergravity in (2 + 1)dimensions hold true, making it possible to analyze its canonicalstructure using the Dirac constraint formalism [5].

In this model, we consider an �(3) gauge field A�a with a topolog-

ical action that couples to a Majorana field �a with spin-3/2. The

Lagrangian is given by

� � �1

2���

a Fa�� �1

2���� �

a��D�ab�5

b (10)

where

F��a � ��A�

a � ��A�a � �abcA�

bA�c (11)

and

D�ab � �� ab � �apbA�

p (12)

The field ���a in (10) is a Lagrange multiplier field, much like the

field e�i in (6). In the next section, we obtain the constraints asso-

ciated with this model, and from the first-class constraints obtainthe symmetries that leave the action following from � in (10)invariant. These constraints are both bosonic and fermionic andsatisfy a closed algebra that is distinct from the algebra of con-straints in supergravity.

3. Canonical analysisThe Lagrangian of (10) can be rewritten

� � �1

2�ij

aFija � �0i

a F0ia � �ijk�0

a�iDjab�5k

b �1

2i

a�0Djab�5k

b

�1

2i

a�jD0ab�5k

b� (13)

It is now apparent that the momenta that conjugate to �ija,

�0ia , A0

a, Aia, 0, and i are, respectively,

Pija � Pi

a � pa � 0 (14a)

pia � �0i

a (14b)

�0a � 0 (14c)

�ia � �

1

2�ijkj�k�5 (14d)

Equations (14a) and (14b) constitute a set of bosonic second-classconstraints

�1ia � Pi

a (15a)

�2ia � pi

a � �0ia (15b)

such that we have the Poisson bracket

��1ia , �2j

a � � ab ij (16)

So also, the constraints of (14f) are primary second-class as wellbecause

�ia � �i

a �1

2�ijkj�k�5 (17)

satisfies

��iaT, �j

b� � ��ijkC�k�5 (18)

From (16) and (18) we find the DB

�A, B�∗ � �A, B� � ��A, �2ia ���1i

a , B� � �A, �1ia ���2i

a , B�

� �A, �ia�� i2�j�i�0C���j

a, B�� (19)

From this DB we see that

�ia,j

b�� �i

2�j�i�0

ab (20)

The canonical Hamiltonian that follows from � in (13) is

�c � iaT�i

aT � Aiapi

a � �

�1

2�ij

aFija � �ijk� 12i

a�0Djab�5k

b � 0a�iDj

ab�5kb�

� A0a�Di

abpib �

1

2�ijk�

abcib�j�5k

c� (21)

The primary constraints that are not second-class give rise to thefollowing secondary constraints:

�Pija,�c�� � �1

2Fija � �

1

2�ij

a (22)

�p0a,�0�� � �Diabpi

b �1

2�ijk�

abcib�j�5k

c� � �a (23)

��0a,�c�� � �C(�ijk�iDj

ab�5kb) � �C� � �C�c (24)

Not all of the constraints �ija are independent, as on account of the

Bianchi identity they are related by

�ijkDiab�jk

b � 0 (25)

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It is now possible to establish the following DB algebra:

��a,�b�∗ � �abc�c (26a)

��a,�b�∗�

1

2�ijk�apb�ij

p�k�5 (26b)

��a,�b�∗ � �abc�c (26c)

��ija,�b� � �abc�ij

a (26d)

with all other DB involving �a, �ija, and �a vanishing.

The Hamiltonian of (21) can be rewritten as

�c �1

2�ij

a�ija � 0

a�a � A0a�a � �x (27)

where the “extra” part of �c is

�x �1

2�ijki

a�0Djab�5k

b (28)

We find that

��a, � d3x �x�∗� �

1

2�abc�ijk�ij

b�0�5kc (29a)

and

��a, � d3�x�∗� ��ij

a, � d3�x�∗� 0 (29b)

and so by (26) and (29) we find that (Pija, p0

a, �0a) are primary first-

class constraints, (�ija, �a, �a) are the corresponding secondary

first-class constraints, and there are no further first-class con-straints.

These first-class constraints lead to a generator of gauge trans-formations that is of the form

G � aijaPij

a � bap0a � �0

aca � �ija�ij

a � �a�a � �a�a (30)

By HTZ, the coefficients of the secondary constraints �ija, �a, and

�a are taken to be field independent. The coefficients of theprimary constraints then follow from the HTZ equation [3]. Wefind that

aija � 2D0

ab�ijb � �abc[�b�ij

c � �b�ijk(�k�50c � �0�5k

c)] � 0 (31a)

ba � �D0ab�b (31b)

ca � �D0ab�b � �abc�b0

c (31c)

so that

G � Pija�2D0

ab�ijb � �abc[�b�ij

c � �ijk�b(�k�50

c � �0�5kc)]� � p0

aD0ab�b

� �0a(D0

ab�b � �abc�b0c ) � �ij

a�ija � �a�a � �a�a (32)

One can now compute the change in a dynamical variable X gen-erated by G, X � �X, G�∗. We find that

A�a � D�

ab�b (33a)

�a � �D�

ab�b � �abc�b�c (33b)

���a � ����D

ab��b � �����abcb���5�c (33c)

where

�ija � �

1

2�ijk�k

a �0a � 0 (34)

The gauge transformation associated with the gauge parameter�a is an ordinary non-abelian gauge transformation. The one as-sociated with �a holds as a result of the Bianchi identity. Thefermionic gauge symmetry associated with the gauge parameter�a mixes the bosonic field ���

a with the fermionic field �a . None of

these transformations are associated with space–time symmetryand hence this is not a supergravity model.

Finally, if GI is the generator associated with the gauge func-tions (�I ij

a ,�Ia,�I

a), then by computing �GI,GJ�� we see that the DB of

the generators GI and GJ gives rise to a generator GK with

�Kija � �

1

2�ijk�

abc�Ib�k�5�J

c � �abc(�Ib�Jij

c � �Iijb �J

c) (35a)

�Ka � �abc�I

b�Jc (35b)

�Ka � �abc(�I

b�Ja � �J

b�Ia) (35c)

We thus have a closed algebra for the bosonic and fermionicgauge transformations that leaves the action of (10) invariant. Noauxiliary fields are required.

In this model, there are 20 bosonic degrees freedom in phasespace (A�

a , ���a , and their conjugate momenta) plus 16 fermionic

degrees of freedom (�a and its conjugate momentum). From (14a)

and (14b) we have six bosonic second-class constraints and from(14d), six fermionic second-class constraints. By (14a) there are fourprimary first-class bosonic constraints; in addition there are threesecondary first-class bosonic constraints (by (22), (23), and (25)).The four first-class fermionic constraints are given by (14c) (pri-mary) and (24) (secondary).When one accompanies each first-classconstraint with an appropriate gauge condition, we see that of the36 degrees of freedom present, only two are left in phase spacebecause of the 34 constraints present. This single degree of free-dom is fermionic.

3. DiscussionThe Dirac constraint formalism [5], when accompanied by the

HTZ procedure for deriving the generator of gauge transforma-tions from the first-class constraints in a theory [3], has proven tobe a useful way of analyzing both bosonic and fermionic gaugesymmetries. This has been demonstrated both by considering thespinning particle [6], and supergravity in (2 + 1) dimensions [4].Although it has not yet been feasible to apply this analysis tosupergravity in (3 + 1) dimensions, we have shown it to be possibleto adapt the treatment of supergravity in (2 + 1) dimensions toformulate a (3 + 1)-dimensional model in which the bosonic andfermionic gauge transformations form a closed algebra that is notthat of supergravity. Quantization can proceed in the way out-lined in ref. 4; this will result in both bosonic and fermionic“ghosts.”

AcknowledgmentsR. Macleod had useful suggestions.

McKeon 21

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References1. J. Wess and J. Bagger. Supersymmetry and supergravity. Princeton University

Press, Princeton, N.J. 1983.2. P.D. Alvarez, M. Valenzuela, and J. Zanelli. J. High Energy Phys. 1204, 058

(2012). doi:10.1007/JHEP04(2012)058.3. M. Henneaux, C. Teitelboim, and J. Zanelli. Nucl. Phys. B, 332, 169 (1990).

doi:10.1016/0550-3213(90)90034-B.4. D.G.C. McKeon. arXiv:1203.6046. 2012.5. P.A.M. Dirac. Lectures on quantum mechanics. Dover Publications, Mineola,

N.Y. 2001.6. D.G.C. McKeon. Can. J. Phys. 90, 701 (2012). doi:10.1139/p2012-076.7. S. Deser, J.H. Kay, and K.S. Stelle. Phys. Rev. D, 16, 2448 (1977). doi:10.1103/

PhysRevD.16.2448.8. M. Pilati. Nucl. Phys. B, 132, 138 (1977). doi:10.1016/0550-3213(78)90262-6.9. E.S. Fradkin and M. Vasiliev. Phys. Lett. B, 72, 70 (1977). doi:10.1016/0370-

2693(77)90065-X.

Appendix A: Conventions and notationWe employ the metric g�� � diag(����) with �0123 �

�1 � �123 � �123. For Dirac matrices,

�0 � �0 � �0 11 0 � �i � ��i � �0 �i

i 0 � �� �1

4i[��, ��]

where i is a Pauli spin matrix. If �5 � �5 � i�0�1�2�3 then

������ � ��g�� � ��g�� � ��g�� � i��������5

and

�i�j � � ij � i�ijk�k�0�5, �ijk�j�ℓ�k � �2i iℓ�0�5

The “charge conjugation” matrix C � �CT � C�1 � �0�2 satisfiesC��C

�1 � ���T. Furthermore we have �0���0 � ��

†.A spinor is Majorana if � c � CT where � � †�0. This

implies that � �TC. If � is also a Majorana spinor, then � �(�)† � � with both � and being Grassmann.

We use the left derivative for Grassmann variables �a so that

d

d�a

(�b�c) � ab�c � ac�b

d

dtF(�(t)) � �(t)F ′(�(t))

If (qi,pi � �L/�qi) and (i,�i � �L/�i) are bosonic and fermioniccanonical variables respectively, then for Poisson brackets wehave

�B1, B2� � (B1,qB2,p � B2,qB1,p) � (B1,B2,� � B2,B1,�)

�B, F� � (B,qF,p � F,qB,p) � (B,F,� � F,B,�) � ��F, B�

�F1, F2� � (F1,qF2,p � F2,qF1,p) � (F1,F2,� � F2,F1,�)

The Hamiltonian is

H � qipi � i�i � L

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