ARTICLE

On the gauge symmetries of the spinning particleN. Kiriushcheva, S.V. Kuzmin, and D.G.C. McKeon

Abstract: We reconsider the gauge symmetries of the spinning particle by a direct examination of the Lagrangian using asystematic procedure based on the Noether identities. It proves possible to find a set of local bosonic and fermionic gaugetransformations that have a simple gauge group structure, which is a true Lie algebra, both for the massless and massive case.This new fermionic gauge transformation of the “position” and “spin” variables in the action decouples from that ofthe “einbein” and “gravitino”. It is also possible to redefine the fields so that this simple algebra of commutators of the gaugetransformations can be derived directly starting from the Lagrangian written in these new variables. We discuss a possibleextension of our analysis of this simple model to more complicated cases.

PACS Nos.: 04.20.Fy, 11.10.Ef.

Résumé : Nous reconsidérons les symétries de jauge d’une particule avec spin par un examen direct du Lagrangien en utilisantune procédure systématique basée sur les identités de Noether. Il s’avère possible de trouver un ensemble de transformations dejauge bosoniques et fermioniques locales qui ont une structure de groupe de jauge simple, ce qui est une vraie algèbre de Lie, ala fois pour les cas avec et sans masse. Cette nouvelle transformation de jauge fermionique des variables de « position » et de« spin » dans l’action découple de celle de l’« einbein » et du « gravitino ». Il est aussi possible de redéfinir ces champs de telle sorteque cette algèbre simple de commutateurs des transformations de jauge peut être obtenue directement en partant duLagrangien écrit a l’aide de ces nouvelles variables. Nous discutons une extension possible de notre analyse de ce modèle simplevers des cas plus compliqués. [Traduit par la Rédaction]

1. IntroductionFor four decades, supersymmetry has been studied intensively.

The local version of this symmetry, supergravity, is most easilyrealized by the spinning particle [1–3]; this is supergravity theoryin 0 + 1 dimensions.

In the original presentation of the action for the spinning par-ticle, a particular set of local bosonic and fermionic gauge trans-formations was given [1, 2]; their form appears to be motivated bythe supersymmetric and diffeomorphism gauge transformationspresent in the supergravity action in 3 + 1 dimensions [4]. How-ever, as was noted in [2], these gauge transformations do not havea gauge group structure in which the structure functions are fieldindependent.

We wish in this paper to point out that this deficiency can beovercome by altering the form of the gauge transformations ina simple way. This is systematically done by direct derivationstarting from the Lagrangian. A general form of an arbitrarygauge transformation can be derived from differential identi-ties (DIs), which are linear combinations of Euler–Lagrangederivatives (ELDs) of the action; this method can be applied toany action with a known gauge transformation to search for aform of the local gauge transformations that simplifies thegauge group properties.

This general expression for a gauge transformation obtainedfrom a DI of the action for the spinning particle can also be usedto find a reparametrization of the fields so that the fermionicgauge transformation decouples the “position” and “spin” fieldsfrom the “einbein” and “gravitino” fields.

We note that the gauge symmetry structure of the spinningparticle action can also be studied using the canonical structure of

the action; a generator of both bosonic and fermionic gauge trans-formations that have a simple gauge group structure can be de-rived from first class constraints [5]. The same procedure can beapplied to the superparticle action [6].

2. The spinning particleWe start by examining the general case of a particle action

S � � d�L(qi(�), qi(�)) (1)

where qi��� � �qi/��, and considering a variation of each of thefields qi(�) so that

�S � � d��i

�qi(�) Eqi(2)

In (2), the ELDs Eqiare given by

Eqi�

�L

�qi

� ��L

�qi

(3)

where ∂ � ∂/∂�. If this were to vanish for arbitrary �qi, then wehave the fields qi satisfying the equation of motion Eqi

� 0. How-ever, if the form of �qi is such that �S = 0 for arbitrary qi(�), then wehave a local gauge symmetry of action S. According to the Noether

Received 6 July 2013. Accepted 31 October 2013.

N. Kiriushcheva. The Department of Applied Mathematics, The University of Western Ontario, London, ON N6A 5B7, Canada.S.V. Kuzmin. The Department of Economics, Business, and Mathematics, The King’s University College, London, ON N6A 2M3, Canada.D.G.C. McKeon. The Department of Applied Mathematics, The University of Western Ontario, London, ON N6A 5B7, Canada; The Department ofEconomics, Business, and Mathematics, The King’s University College, London, ON N6A 2M3, Canada.Corresponding author: D.G.C. McKeon (e-mail: dgmckeo2@uwo.ca).

411

Can. J. Phys. 92: 411–414 (2014) dx.doi.org/10.1139/cjp-2013-0351 Published at www.nrcresearchpress.com/cjp on 18 November 2013.

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

San

Fran

cisc

o (U

CSF

) on

05/

07/1

4Fo

r pe

rson

al u

se o

nly.

theorem [7, 8], any gauge symmetry qi ¡ qi + �qi of S satisfies theequation

�i

�qiEqi� 0 (4)

which leads to the differential identity, I,

I� � �i

�qi(�) Eqi(5)

where � is a gauge parameter.The Lagrangian we are particularly concerned with is that of the

spinning particle

L �1

2�e�1�� � i � ie�1��� (6)

where ���� and e(�) are two bosonic fields and (�) and �(�) aretwo fermionic fields, or two pairs of superpartners (e, �) and��, �. The index in � and is raised and lowered by theMinkowski metric tensor.

In refs. 1 and 2, the bosonic gauge invariance present in thisaction is given by the “diffeomorphism” transformation

�fe � �(fe) �f� � �(f�) �f� � f�(�) �f

� f�() (7)

while the fermionic or supersymmetry transformation was cho-sen by the authors of [1, 2] as

��� � i� ��e � i�� ��� � 2�

�� � �e�1� �i

2� (8)

with f(�) and �(�) being bosonic and fermionic gauge parameters,respectively.

Under the transformations of (7) and (8) we find

� fL � �(fL) (9)

��L � � i

2e�� (10)

respectively.The group property of the bosonic transformation of (7) can be

combined into one equation

[� f1, � f2

](�, , e, �) � � f(�, , e, �) (11)

where

f � f2f1 � f1f2 (12)

and so the structure functions of this transformation are fieldindependent. The same is valid also for the commutator ofbosonic and fermionic transformations

[� f, ��](�, , e, �) � ��(�, , e, �) (13)

with the parameter

� � � f� (14)

However, when commuting two fermionic gauge transformationswe find from (7) and (8)

[��1, ��2

](�, , e, �) � � f (�, , e, �) � �� (�, , e, �) (15)

where

f � �2i�1�2

eand � � �

1

2f� (16)

We thus see that the transformations of (7) and (8) do not possessa simple group property, as was noticed by the authors of ref. 2,because of the explicit dependence of the gauge parameters in (16)on the fields.

We will now exploit (4), using the Noether identities, to find aform of gauge transformations that has a simpler gauge group.This method was outlined in refs. 9 and 10. We begin from theELDs of the Lagrangian (6)

Ee ��L

�e� �

1

2e2��� � i��� (17)

E� ��L

��� �

i

2e� (18)

E� ��L

��� ��e�1� �

i

2e� (19)

E ��L

�� �i �

i

2e�� (20)

and so by (7) and (8) we have the bosonic DI

I � �e�Ee � ��E� � �(�) E� � �() E � 0 (21)

and the fermionic DI

� �2�E� � i�Ee � iE� � � �i

2�e�1E � 0 (22)

respectively. We now use the fact that any linear combination ofthese two DIs yields a DI; the number of linearly independent DIscannot be changed [7, 8]. The simplest modification is to multiply(22) by a function h(e), if we want to preserve its tensorial andfermionic nature; the resulting DI corresponds to the gauge trans-formations

��� � �(2�h(e)) ��e � i��h(e) (23)

��� � i�h(e) �� � �e�1� �i

2�h(e) (24)

which gives the commutator

[��1, ��2

]� � ��4i�1�2

dh(e)

deh(e)� (25)

All field dependence in this commutator disappears if we take

412 Can. J. Phys. Vol. 92, 2014

Published by NRC Research Press

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

San

Fran

cisc

o (U

CSF

) on

05/

07/1

4Fo

r pe

rson

al u

se o

nly.

h(e) � �e (26)

so that the fermionic gauge transformation becomes

��� � i�e� ��e � i�e�� ��� � 2�(�e�)

�� � �1

�e� �

i

2� (27)

The commutator of two new fermionic gauge transformations isnow becomes

[��1, ��2

] (�, , e, �) � �f(�, , e, �) (28)

with

f � �2i�1�2 (29)

while together the fermionic and bosonic gauge transformationsresult in

[� f, ��](�, , e, �) � ��(�, , e, �) (30)

with

� � f� �1

2�f (31)

With the gauge transformations of (11) and (27) we see that wehave a gauge algebra whose structure functions are independentof fields; this simple algebra automatically satisfies the Jacobiidentities.

One can supplement the Lagrangian of (6) with a “mass”, or“cosmological” term

L5 �1

2�m2e � i55 � im5�� (32)

The Lagrangian L5 of (32) is invariant under the gauge transforma-tions of (7) and (8) provided we also have [1]

� f5 � f5 (33)

and

��5 � m� (34)

which result in

� fL5 � �(fL5) ��L5 � ��i

2m5 (35)

and the commutator of two fermionic transformations is verysimple

[��1, ��2

]5 � 0 (36)

However, the transformation of (34) has been supplemented by anextra piece given in [2, 3] so that

��5 � m� �i

me�55 �

1

2m� (37)

in order that the gauge algebra of (15) is satisfied.The ELD associated with 5 is

E5� i5 �

i

2m� (38)

when this is combined with the fermionic gauge transformationof (37) we end up with the DI

⇒ � �m �i

me55 �

1

2m� E5

� 0 (39)

where is given by (22) with new “einbein” and “gravitino” ELDsare

Ee ⇒ Ee �1

2m2 (40)

E� ⇒ E� �i

2m5 (41)

We would like to note that, despite of consistency with the gaugealgebra of (15), the gauge transformation (37) with the extra pieceis not legitimate because it introduces a term in the DI propor-tional to the square of ELD E5

, as (39) can be written in the form

� mE5�

1

me5�E5�2 � 0 (42)

It contradicts the definition of a DI as being a linear combinationof ELDs.

We now make use of the function h(e) of (26), which modifiesthe DI so that we have the fermionic gauge transformation

��5 � �em� �i

m�e�55 �

1

2m� (43)

This is consistent with the gauge algebra given by (28)–(31).One can also make an invertible change of variables in the

original action without destroying the gauge algebra. For exam-ple, in ref. 1 a rescaling of fermionic variables

�1

�e � �

1

�e� (44)

5 � �e5 (45)

leads to the actions of (6) and (32) being replaced by

L �1

2�e�1�� � ie�1�() � ie�2��� (46)

and

L5 �1

2�m2e � ie5�(5) � im5�� (47)

Kiriushcheva et al. 413

Published by NRC Research Press

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

San

Fran

cisc

o (U

CSF

) on

05/

07/1

4Fo

r pe

rson

al u

se o

nly.

One can easily obtain the DI in terms of these new variables. Theresulting gauge transformations for these new variables are

� f� � �(f�) �1

2f� � f

� f�() �1

2f

� f5 � f�(5) �1

2f5

(48)

for the bosonic case (the gauge transformations �fe and �f� re-

mains the same as in (7)), and

��e � i�� ��� � 2�e � �e (49)

��� � i� �� � �� (50)

��5 � m� �i

m�5�(5) (51)

for the fermionic case. The fermionic transformations of (49)–(51)are much simpler than those of (8) and (43). In addition, the trans-formations of � and (the “position” and “spin” fields) decouplefrom those of e and � (the “einbein” and “gravitino” fields), as wellas 5 transforms separately from other fields.

Despite this new form of the gauge transformations, we retainthe simple gauge algebra of (11), (12), and (28)–(31) in which allstructure functions are field independent

[� f1, � f2

]field � � ffield with f � f2f1 � f1 f2 (52)

[��1, ��2

]field � �ffield with f � �2i�1�2 (53)

[� f, ��]field � ��field with � � f� �1

2�f (54)

The Jacobi identities automatically hold for such gauge transfor-mations.

A particularly simple form of the gauge transformations for thefields, which almost trivializes the calculations of the gauge alge-bra can be obtained by replacing e by g and � by �== where

e � exp(g) (55)

� ′′ � exp(�g)� (56)

We now find the bosonic transformation

�fg � f � fg (57)

� f�′′ � f� ′′ �

1

2f� ′′ (58)

and the fermionic one

�� � �� ��� � i� (59)

��g � i�� ′′ ��� ′′ � 2� � �g (60)

This last parametrization (55) and (56) has especially simpletransformations that makes calculation of commutators of twosupersymmetry transformations almost trivial.

3. DiscussionBy working directly with the DIs obtained from the action for

the spinning particle, we have derived a set of bosonic (B) andfermionic (F) gauge transformations that have a simple gaugealgebra of the form

[B, B] � B [F, F] � B [F, B] � F (61)

In this algebra, all structure functions are field independent andthe Jacobi identity is satisfied. This is an improvement over theoriginal set of gauge transformations appearing in refs. 1, 2. Note,if we are seeking for gauge transformations of bosonic and fermi-onic fields that form a Lie algebra with field independent struc-ture functions, then the only form possible is that of (61).

The actual form of the gauge transformations has been simpli-fied through a field redefinition while retaining the simple alge-bra of (61) for the gauge transformations.

We would like to investigate more complicated models thathave a local fermionic symmetry to see if similar simplificationscan be effected. An O(N) generalization of the spinning particle,the spinning string and supergravity in D ≥ 3 dimensions shouldall be examined with this objective in mind.

A general problem would be to establish the relationship be-tween the DI of (4) that is satisfied by any gauge transformationand the gauge generator obtained from the first class constraintsarising from the canonical structure of the theory [11, 12].

The gauge generator derived from the first class constraints canbe used to determine a gauge invariance of a theory (even onepreviously unsuspected, as in the case of the first order Einstein–Hilbert action in two dimensions [13]). However, it has not asyet proven possible to determine directly from the Lagrangianand its ELDs all independent DIs of the form (4) and conse-quently, all gauge symmetries, though once one has a gaugetransformation, alternate gauge transformations can easily befound by using this DI.

AcknowledgementR. Macleod provided a helpful suggestion.

References1. L. Brink, S. Deser, B. Zumino, P. Di Vecchia, and P. Howe. Phys. Lett. B, 64, 435

(1976). doi:10.1016/0370-2693(76)90115-5.2. L. Brink, P. Di Vecchia, and P. Howe. Nucl. Phys. B, 118, 76 (1977). doi:10.1016/

0550-3213(77)90364-9.3. Erratum. Phys. Lett. B, 68, 488 (1977).4. P.van Niewenhuizen. Phys. Rep. 68, 189 (1981). doi:10.1016/0370-1573(81)

90157-5.5. D.G.C. McKeon. Can. J. Phys. 90, 701 (2012). doi:10.1139/p2012-076.6. D.G.C. McKeon. arXiv:1209.4909 [hep-th].7. E. Noether. Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math.-phys.

Klasse, 235. 1918.8. E. Noether. (M.A. Tavel’s English translation): arXiv:physics/0503066.9. N. Kiriushcheva and S.V. Kuzmin. Gen. Rel. Grav. 42, 2613 (2010). doi:10.1007/

s10714-010-1003-7.10. N. Kiriushcheva, P.G. Komorowski, and S.V. Kuzmin. arXiv:1112.5637 [hep-th].11. L. Castellani. Ann. Phys. 143, 357 (1982). doi:10.1016/0003-4916(82)90031-8.12. M. Henneaux, C. Teitelboim, and J. Zanelli. Nucl. Phys. B, 332, 169 (1990).

doi:10.1016/0550-3213(90)90034-B.13. N. Kiriushcheva, S.V. Kuzmin, and D.G.C. McKeon. Mod. Phys. Lett. A, 20,

1895 (2005). doi:10.1142/S0217732305018086.

414 Can. J. Phys. Vol. 92, 2014

Published by NRC Research Press

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

San

Fran

cisc

o (U

CSF

) on

05/

07/1

4Fo

r pe

rson

al u

se o

nly.