fulltext

The Maxwell field with the inclusion of fermions on a lattice

and Itô terms

I. A. MUN˜OZMAYA(1), J. M. RIVERAREBOLLEDO(2)(*) and M. SOCOLOVSKY(3)

(1) Departamento de Metal-Mecánica, Instituto Tecnológico de Veracruz Apdo. Postal 539, 91860, Veracruz, Ver. México

(2) Departamento de Física, Escuela Superior de Física y Matemáticas del Instituto Politécnico Nacional - U. P. Zacatenco, 07738, México, D.F. (3) Instituto de Ciencias Nucleares, Universidad Autónoma de México

Circuito exterior, A. P. 70543, 04510 México, D.F. (ricevuto il 9 Agosto 1996; approvato il 4 Marzo 1997)

Summary. — In this note we have searched for nonsymmetric Itô terms when the

Maxwell field on a lattice includes fermions. We show explicitly that the Itô terms are not present in the corresponding effective continuum action if the space-time dimension of the lattice d is equal to 2 and 3. This is an expected result due to the flatness of the U(1) manifold.

PACS 03.20 – Classical mechanics of discrete systems: general mathematical aspects.

As is known the so-called Itô terms appear in lattice gauge theories when the quantum continuum action is calculated. In fact, one looses gauge invariance in the naive continuum limit of the action, as, for instance [1, 2], in the SU(2) gauge group and s-model cases. This is due to the nonzero curvature of the corresponding manifold. That is not the case for the U(1) group, where the manifold is flat. The Itô terms have to be aggregated to the action in order to restore gauge invariance.

On the other hand, it has been shown [3] that the Itô terms do not appear in the effective continuum action for pure Maxwell field provided the space-time dimension d is E4. The result is expected because of the flatness of the U(1) manifold. Nevertheless, as the authors of ref. [3] pointed out, such result is not necessarily the same when the presence of fermions is considered. Following this idea, we have included fermions in the lattice under the interaction of the electromagnetic field. We wish to prove explicitly that the related partition function is gauge invariant so that the Itô terms are absent. We have done it for d G3. In fig. 1 the cubic lattice is drawn with its sites and in fig. 2 the main plaquettes of fig. 1 are displayed. a1 and a2 are the plaquettes for the

case d 42.

(*) With support from COFAA-IPN, México.

Fig. 1. – Three-dimensional lattice (the numbers represent the sites).

The complete action for d-dimensional compact electrodynamics can be written as [4]

(1) _{S 4}

!

n , mc – n]K

!

m [ (I 1gm) U *mnd d n 1m, m1 (I 2 gm) Ummddn 2m, m] 1ddn , m( cm1 1(Ma) d 24 g2

!

m , n[I 2Re (UmnUn , n 1mU *m , n 1nU *nn) ] ,

mass and a is the lattice spacing. Also n (or m) represents a lattice site, and m and n stand for the lattice directions. The hopping parameter K is given by [5]

K 4 1 2 M(a) Q a 28 Ka K0 2 1 8 , (2) whereas dd n , m is defined as dd n , m4 a



2pOa pOa dd q ( 2 p)de 2iaqm(n 2m)m_. (3)

Umn is in turn defined in terms of the link variable umn (see fig. 2) through Umn4 liumn

(4)

and satisfies [5] U_{2m , n 1 m4 U}21

mn , so that umn4 2 u2m , n 1 m. The associated partition

function can be expressed as [6]

Z 4

u

»

l



2p p dul

v

( det Anm) exp

y

2 (Ma)d 24 g2 P 41

!

2d 21 ( 1 2cos uP)

z

, (5)

where l means the link (n , m) and P represents the plaquette. On the other hand, the periodicity of the lattice tells us that dd

n 1m, m4 ddn 2m, m. With

the help of this relation equation and referring to fig. 2, we have

.

`

/

`

´

uP1fa14 u11 b1, uP34 a34 u11 b3, uP24 a24 2 a1, uP44 a44 2 a3, (6) with

.

/

´

u14 u22, b14 u112 u142 u21, b34 u332 u262 u32. (7)

Further, one finds for the other plaquette variables:

.

`

/

`

´

a54 u252 u152 u262 u17, a64 u211 u342 u252 u31, a74 u111 u322 u152 u31, a84 u141 u331 u172 u34. (8)

The matrix elements Anm are Anm4 K

!

m [ ( 1 1gm) U *mnd d n 1m, m1 ( 1 2 gm) Ummddn 2m, m] 1ddn , m. (9)

The corresponding determinant reduces, for d 42, to

D24

N

N

N

1 2 Keiu11 0 2 Keiu21 2 Ke2iu11 1 2 Keiu22 0 0 2 Ke2iu22 1 2 Ke2iu14 2 Ke2iu21 0 2 Keiu14 1

N

N

N

4 1 2 16 K21 32 K4( 1 2cos a1) , (10)

while for d 43 one gets (11) D34 4

N

N

N

1 2 Keiu11 0 2 Keiu21 2 Keiu31 0 0 0 2 Ke2iu11 1 2 Keiu22 0 0 2 Keiu32 0 0 0 2 Ke2iu22 1 2 Ke2iu14 0 0 2 Keiu33 0 2 Ke2iu21 0 2 Keiu14 1 0 0 0 2 Keiu34 2 Ke2iu31 0 0 0 1 2 Keiu15 0 2 Keiu25 0 2 Ke2iu32 0 0 2 Ke2iu15 1 2 Keiu26 0 0 0 2 Ke2iu33 0 0 2 Ke2iu26 0 2 Keiu17 0 0 0 2 Ke2iu34 2 Ke2iu25 0 2 Ke2iu17 1

N

N

N

4 4 32 K4(21116K2 2 32 K4) cos a11 32 K4(21112K22 16 K4) cos a31

11024 K8( cos a1cos a51 cos a3cos a6) 2128K6( 1 24K2) Q

Q [ cos (a11a6)1cos (a12a7)1cos (a11a8)1cos (a32a5)1cos (a32a7)1cos (a32a8)]2

2128 K6

(

( cos (a11 a62 a7) 1cos (a11 a61 a8)

)

2

2512 K8

(

cos (a11 a51 a6) 1cos (a11 a52 a7) 1cos (a11 a51 a8) 1

1cos (a12 a71 a8)

)

1 D8 ,

where D8 does not depend on a1 and a3.

We notice that the basic difference of the partition function, eq. (5), from eq. (6) of ref. [3], is the factor (included in the integration) det Anm. The square bracket was

shown there to be gauge invariant for d E4. Then, inserting eq. (10) into eq. (5) we obtain, for d 42, Z 4

o

p K 8

g

1 216K 2 1 1 K 8 Q 8 K 4

h

»

l 8



_du l 8

»

P 8 exp [2K 8(12cos uP 8 ) ] , (12)

(13) _{Z 4}

»

l 8



_{dul 8}

_»

P 8D9Q exp [2K 8(12cos uP 8) ] Q



2Q Q

du1D(u1) exp [2K 8(a211 a23) ] 4

4

»

l 8



_du l 8

»

P 8 D8Q exp [2K 8(12cos uP 8 ) ] Q exp

k

₂1 2 K 8(b12 b3) 2

l

Q Q



2Q Q du 81D(u 81) exp [22K 8 u182] .

Taking cos (a11 a62 a7) as a typical term of D3 and using the continuum limits

cos uP ` a K01 2 1 2u 2 P, sin uP ` a K0uP,

the corresponding contribution is

(14) _{I B} 1 8 K 8

o

p 2 K 8 Q

k

exp

k

2 1 2K 8(b12 b3) 2

ll

Q Q ]2114(b12 b3) sin (a62 a7) 1 [82 (b12 b3)2] cos (a62 a7)( .

Since I depends essentially on b12 b3

(

with sin (a62 a7) and cos (a62 a7) absorbed

into

!

_{P 8}

)

, it is easy to convince oneself that b12 b3, as defined by (7), is invariant under

the transformation

umnK umn1 g[L(n 1 m) 2 L(n) ] .

The other integrals that appear in (13), can be treated in the same fashion and again we can prove that they are gauge invariant. Therefore this property is fulfilled by the complete partition function

(

of course, for d 42, eq. (12), this result is trivial

)

. In this way we arrive at the desired result: no Itô terms are present in the effective continuum action of lattice gauge theories when the Maxwell field includes fermions. We have shown this to be true for lattice dimensions d 42 and d43.

R E F E R E N C E S

[1] PATRASCIOIU A. and RICHARD J. L., Lett. Math. Phys. 9 (1985) 191. [2] PATRASCIOIU A., Phys. Rev. Lett., 54 (1985) 1102.

[3] D’OLIVO J. C. and SOCOLOVSKY M., Nuovo Cimento B, 101 (1988) 177. [4] WILSON K. G., Phys. Rev. D, 10 (1974) 2445.

[5] CHAICHIAN M. and NEHIPA N. F., Introduction to Gauge Field Theories, Texts and Monograps in Physics (Springer-Verlag, Berlin) 1984.