Local supersymmetry leads to boundary conditions for fermionic fields in
one-loop quantum cosmology involving the Euclidean normal to the boundary and a
pair of independent spinor fields. This paper studies the corresponding
classical properties, i.e. the classical boundary-value problem and boundary
terms in the variational problem. Interestingly, a link is found with the
classical boundary-value problem when spectral boundary conditions are imposed
on a 3-sphere in the massless case. Moreover, the boundary term in the action
functional is derived.Comment: 8 pages, plain-tex, recently appearing in Foundations of Physics
  Letters, volume 7, pages 303-308, year 199

Esposito, Giampiero

Morales-Tecotl, Hugo A.

Pollifrone, Giuseppe

English

arXiv

Local supersymmetry leads to boundary conditions for fermionic fields in one-loop quantum cosmology involving the Euclidean normal to the boundary and a pair of independent spinor fields. This paper studies the corresponding classical properties, i.e. the classical boundary-value problem and boundary terms in the variational problem. Interestingly, a link is found with the classical boundary-value problem when spectral boundary conditions are imposed on a 3-sphere in the massless case. Moreover, the boundary term in the action functional is derived

Esposito, G

Morales-Tecotl, H A

Pollifrone, G

CERN Document Server

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August 1993, SISSA Ref. 137/93/A, DSF preprint 93/27
BOUNDARY TERMS FOR MASSLESS
FERMIONIC FIELDS
Giampiero Esposito
1;2
, Hugo Morales-Tecotl
3
and Giuseppe Pollifrone
1
1
Istituto Nazionale di Fisica Nucleare
Mostra d'Oltremare Padiglione 20, 80125 Napoli, Italy;
2
Dipartimento di Scienze Fisiche
Mostra d'Oltremare Padiglione 19, 80125 Napoli, Italy;
3
Scuola Internazionale Superiore di Studi Avanzati
Via Beirut 2-4, 34013 Trieste, Italy.
Abstract. Local supersymmetry leads to boundary conditions for fermionic elds in
one-loop quantum cosmology involving the Euclidean normal
e
n
A
0
A
to the boundary and
a pair of independent spinor elds  
A
and
e
 
A
0
. This paper studies the corresponding
classical properties, i.e. the classical boundary-value problem and boundary terms in the
variational problem. If
p
2
e
n
A
0
A
 
A

e
 
A
0
 
A
0
is set to zero on a 3-sphere bounding at
Euclidean 4-space, the modes of the massless spin-
1
2
eld multiplying harmonics having
positive eigenvalues for the intrinsic 3-dimensional Dirac operator on S
3
should vanish on
S
3
. Remarkably, this coincides with the property of the classical boundary-value problem
when spectral boundary conditions are imposed on S
3
in the massless case. Moreover, the
1
Boundary terms for massless fermionic elds
boundary term in the action functional is proportional to the integral on the boundary of

A
0
e
n
AA
0
 
A
.
2
Boundary terms for massless fermionic elds
Locally supersymmetric boundary conditions have been recently studied in quantum cos-
mology to understand its one-loop properties. They involve the normal to the boundary
and the eld for spin
1
2
, the normal to the boundary and the spin-
3
2
potential for gravitinos,
Dirichlet conditions for real scalar elds, magnetic or electric eld for electromagnetism,
mixed boundary conditions for the 4-metric of the gravitational eld (and in particular
Dirichlet conditions on the perturbed 3-metric). The aim of this letter is to describe the
corresponding classical properties in the case of massless spin-
1
2
elds.
For this purpose, we consider at Euclidean 4-space bounded by a 3-sphere of radius
a. The spin-
1
2
eld, represented by a pair of independent spinor elds  
A
and
e
 
A
0
, is
expanded on a family of 3-spheres centred on the origin as [1-3]
 
A
=

 
3
2
2
1
X
n=0
(n+1)(n+2)
X
p=1
(n+1)(n+2)
X
q=1

pq
n
h
m
np
( )
nqA
+ er
np
( )
nqA
i
(1)
e
 
A
0
=

 
3
2
2
1
X
n=0
(n+1)(n+2)
X
p=1
(n+1)(n+2)
X
q=1

pq
n
h
em
np
( )
nqA
0
+ r
np
( )
nqA
0
i
: (2)
With our notation,  is the Euclidean-time coordinate, the 
pq
n
are block-diagonal matrices
with blocks

1 1
1  1

, the   and -harmonics obey the identities described in [1,3]. Last
but not least, the modes m
np
and r
np
are regular at  = 0, whereas the modes em
np
and
er
np
are singular at  = 0 if the spin-
1
2
eld is massless. Bearing in mind that the harmonics

nqA
and 
nqA
0
have positive eigenvalues
1
2

n +
3
2

for the 3-dimensional Dirac operator
on the bounding S
3
[3], the decomposition (1-2) can be re-expressed as
 
A
=  
A
(+)
+  
A
( )
(3)
3
Boundary terms for massless fermionic elds
e
 
A
0
=
e
 
A
0
(+)
+
e
 
A
0
( )
: (4)
In (3-4), the (+) parts correspond to the modes m
np
and r
np
, whereas the ( ) parts
correspond to the singular modes em
np
and er
np
, which multiply harmonics having negative
eigenvalues  
1
2

n +
3
2

for the 3-dimensional Dirac operator on S
3
. If one wants to nd
a classical solution of the Weyl equation which is regular 8 2 [0; a], one is thus forced to
set to zero the modes em
np
and er
np
8 2 [0; a] [1]. This is why, if one requires the local
boundary conditions [3]
p
2
e
n
A
0
A
 
A

e
 
A
0
= 
A
0
on S
3
(5)
such a condition can be expressed as [3]
p
2
e
n
A
0
A
 
A
(+)
= 
A
0
1
on S
3
(6)

e
 
A
0
(+)
= 
A
0
2
on S
3
(7)
where 
A
0
1
and 
A
0
2
are the parts of the spinor eld 
A
0
related to the - and -harmonics
respectively. In particular, if 
A
0
1
= 
A
0
2
= 0 on S
3
as in [2,3], one nds
1
X
n=0
(n+1)(n+2)
X
p=1
(n+1)(n+2)
X
q=1

pq
n
m
np
(a)
e
n
A
0
A

A
nq
= 0 (8)
1
X
n=0
(n+1)(n+2)
X
p=1
(n+1)(n+2)
X
q=1

pq
n
r
np
(a) 
A
0
nq
= 0 (9)
where a is the 3-sphere radius. Since the harmonics appearing in (8-9) are linearly in-
dependent, these relations lead to m
np
(a) = r
np
(a) = 0 8n; p. Remarkably, this simple
4
Boundary terms for massless fermionic elds
calculation shows that the classical boundary-value problems for regular solutions of the
Weyl equation subject to local or spectral conditions on S
3
share the same property pro-
vided 
A
0
is set to zero in (5): the regular modes m
np
and r
np
should vanish on the
bounding S
3
.
To study the corresponding variational problem for a massless fermionic eld, we
should now bear in mind that the spin-
1
2
action functional in a Riemannian 4-geometry
takes the form [2,3]
I
E
=
i
2
Z
M
h
e
 
A
0

r
AA
0
 
A

 

r
AA
0
e
 
A
0

 
A
i
p
det g d
4
x +
b
I
B
: (10)
This action is real, and the factor i occurs by virtue of the convention for Infeld-van der
Waerden symbols used in [2,3]. In (10)
b
I
B
is a suitable boundary term, to be added to en-
sure that I
E
is stationary under the boundary conditions chosen at the various components
of the boundary (e.g. initial and nal surfaces, as in [1]). Of course, the variation I
E
of
I
E
is linear in the variations  
A
and 
e
 
A
0
. Dening  
2
i
and 
b
I
B
 I
B
, variational
rules for anticommuting spinor elds lead to


I
E

=
Z
M
h
2
e
 
A
0

r
AA
0
 
A
i
p
det g d
4
x 
Z
M
h
r
AA
0
e
 
A
0

2 
A
i
p
det g d
4
x
 
Z
@M
h
e
n
AA
0


e
 
A
0

 
A
i
p
det h d
3
x +
Z
@M
h
e
n
AA
0
e
 
A
0

 
A
i
p
det h d
3
x
+ I
B
(11)
where I
B
should be chosen in such a way that its variation I
B
combines with the sum of
the two terms on the second line of (11) so as to specify what is xed on the boundary
5
Boundary terms for massless fermionic elds
(see below). Indeed, setting  = 1 and using the boundary conditions (5) one nds
e
n
AA
0
e
 
A
0
=

p
2
 
A
  
e
n
AA
0

A
0
on S
3
: (12)
Thus, anticommutation rules for spinor elds [1] show that the second line of equation (11)
reads
I
@M
  
Z
@M
h

e
 
A
0

e
n
AA
0
 
A
i
p
det h d
3
x+
Z
@M
h
e
n
AA
0
e
 
A
0

 
A
i
p
det h d
3
x
= 
Z
@M
e
n
AA
0
h

A
0

 
A
  
A
0

 
A
i
p
det h d
3
x : (13)
Now it is clear that setting
I
B
 
Z
@M

A
0
e
n
AA
0
 
A
p
det h d
3
x ; (14)
enables one to specify 
A
0
on the boundary, since

h
I
@M
+ I
B
i
= 2
Z
@M
e
n
AA
0


A
0

 
A
p
det h d
3
x : (15)
Hence the action integral (10) appropriate for our boundary-value problem is
I
E
=
i
2
Z
M
h
e
 
A
0

r
AA
0
 
A

 

r
AA
0
e
 
A
0

 
A
i
p
det g d
4
x
+
i
2
Z
@M

A
0
e
n
AA
0
 
A
p
det h d
3
x : (16)
Note that, by virtue of (5), equation (13) may also be cast in the form
I
@M
=
1
p
2
Z
@M
h
e
 
A
0


A
0

 


e
 
A
0


A
0
i
p
det h d
3
x ; (17)
6
Boundary terms for massless fermionic elds
which implies that an equivalent form of I
B
is
I
B

1
p
2
Z
@M
e
 
A
0

A
0
p
det h d
3
x : (18)
The local boundary conditions studied at the classical level in this paper, have been
applied to one-loop quantum cosmology in [2-4]. Interestingly, our work seems to add
evidence in favour of quantum amplitudes having to respect the properties of the classical
boundary-value problem. In other words, if fermionic elds are massless, their one-loop
properties in the presence of boundaries coincide in the case of spectral [1,3,5] or local
boundary conditions [2-4], while we nd that classical modes for a regular solution of
the Weyl equation obey the same conditions on a 3-sphere boundary with spectral or local
boundary conditions, provided the spinor eld 
A
0
of (5) is set to zero on S
3
. We also hope
that the analysis presented in Eqs. (10)-(18) may clarify the spin-
1
2
variational problem
in the case of local boundary conditions on a 3-sphere (cf. the analysis in [6] for pure
gravity).
Acknowledgments
We are indebted to George Ellis for encouraging our work. The rst author has learned
the two-component spinor analysis of fermionic elds from Peter D'Eath. The second
author thanks for nancial support the Ministero Italiano per l'Universita e la Ricerca
Scientica e Tecnologica, and CONACyT (Reg. No. 55751). Financial support by the
Istituto Nazionale di Fisica Nucleare is gratefully acknowledged from the third author.
7
Boundary terms for massless fermionic elds
REFERENCES
1. P. D. D'Eath and J. J. Halliwell, Phys. Rev. D 35, 1100 (1987).
2. P. D. D'Eath and G. Esposito, Phys. Rev. D 43, 3234 (1991).
3. G. Esposito, Quantum Gravity, Quantum Cosmology and Lorentzian Geometries, Lec-
ture Notes in Physics, New Series m: Monographs, Vol. m12 (Springer-Verlag, Berlin,
1992).
4. A. Y. Kamenshchik and I. V. Mishakov, Phys. Rev. D 47, 1380 (1993).
5. P. D. D'Eath and G. Esposito, Phys. Rev. D 44, 1713 (1991).
6. J. W. York, Found. Phys. 16, 249 (1986).
8


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