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Spin 52 on the lattice
Volume 126B, number 1,2
PHYSICS LETTERS
23 June 1983
SPIN 5/2 ON THE LATTICE ~
C.P. van den DOEL
Department of Physics, University of California, Santa Cruz, CA 95064, USA Received 29 March 1983
We study spin 5/2 fields on the lattice. Species doubling is found to be partially curable with an analogue of the KogutSusskind formalism, reducing the degeneracy from 16 to 4 while preserving local gauge invariance.
In a previous letter [1] we showed that spin 3/2 fields could be put on the lattice with an analogue of the Kogut-Susskind "staggered" fermion formalism [2]. The advantage of this formalism over an analogue of Wilson's method [3] is that local gauge invariance ("supersymmetry") is preserved. However, a spin 3/2 "staggered" fermion field describes 4 particles instead of only 1 (on a four-dimensional euclidean lattice). In this letter we generalize the construction o f "staggered" fermions to spin 5/2 fields. The continuum action for free, massless spin 5/2 fermions is given by [41
The naive lattice version of (1) is obtained by replacing the derivative a~ with the double lattice derivative (using standard notation)
a.f(x) -~ } [ f ( x + p ) - f ( x - p)] -~ Oufin the following.
(5)
The corresponding "naive" lattice action has the same species doubling problem as the naive lattice spin 1/2 [5] and spin 3/2 [l] actions, since the double lattice derivative (5) corresponds to sin pU in momentum space and the sine has 16 low momentum regions, so it would correspond to 16 particles. Starting from eq. (1) (where 3 is now the lattice derivative) we can diagonalize the v-matrices by writing [6]
O) The field ~uv is symmetric in the tensor indices and carries a Dirac index. We shall work in euclidean space and take the 7-matrices to be hermitian. We choose a representation where the ,),i are real and 7 0 is purely imaginary. In this representation,
..7
•
(2)
The action (1) is invariant under the following local gauge transformations
~..(x)
= r(x
+ ¢ + o)(7o) z~'xx × . . ( x ) ,
(6)
where
r(x) = (vo) x o (71)x~ (72) x2 (73)x3.
(7)
The matrices T(x) are unitary and satisfy
T+(x)'I'uT(x + fi) = flu(x),
(8)
with
6 t~uv = a~zXv + avXu ,
(3)
~0(x) = 1,
where ~ is a spin 3/2 field that satisfies 7"7t• = 0 ,
(4)
Supported by a grant from the National Science Foundation.
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
~l(X) = ( - 1 ) ~° ,
772(x) = ( - 1 ) xo+xl ,
r/3(x ) = ( - 1 ) x°+xl+x2 .
(9)
Substituting eq. (6) in the action (1) leads to 87
Volume 126B, number 1,2
S
~
T
PHYSICS LETTERS
The constraint (12) follows directly from eqs. (4) and (13). The action (10) has a continuous U(1) invariance,
,
x,indices
Xuv(x) -* exp[iae(x)]
-- ,Tv(X + ~)%(X + ~)'7~,(X + ~ + 5)GX~,~(X )
e(x) = ( - 1 ) x°+xl +x2 +x3 . •
(lO)
The field Xuv still carries a Dirac index a = 1, ..., 4, but since the action ( I 0 ) is diagonal in Dirac space we can retain only one of the components and by doing so we reduce the doubling from 16 to 4. The local supersymmetry (3), (4) for the field Xuv becomes 6Xuv(x) =
Ou#v(x) + Ovpu(x),
(11)
with
~u(x)pu(x)
=
O.
(12)
The field pu(x) is a "staggered" spin 3]2 field, defined in terms o f the field Xu(x ) as,
Xu (x) = r(x +/~) (3'0) x xx x+ 1 Pu ( x ) .
88
Xuv(x),
(14)
where
+ 2rtv(x +/])3xXxu(x)] + x~a [ ~ % ( x ) G x u u ( x ) - % ( x ) G x u d x ) l
23 June 1983
(13)
(15)
In the continuum limit this U(1) symmetry becomes embedded in the group 0(4). We foresee no problems in extending the above analyses to fields o f arbitrary non-integer spin. This is left as an exercise to the reader.
References [1] G. Maturana and C,P. van den Doel, Phys. Lett. 123B (1983) 332. [2] L. Susskind, Phys. Rev. D16 (1977) 3031. [3] K.G. Wilson, Phys. Rev. D10 (1974) 2445; in: New phenomena in subnuclear physics, Erice Lectures 1975. ed. A. Zichichi (Plenum, New York, 1977). [4] J. Fang and C. Fronsdal, Phys. Rev. D18 (1978) 3630. [5] L.H. Karsten and J. Smit, Nucl. Phys. B183 (1981) 103. [6] N. Kawamoto and J. Smit, Nucl. Phys. B192 (1981) 100.
PHYSICS LETTERS
23 June 1983
SPIN 5/2 ON THE LATTICE ~
C.P. van den DOEL
Department of Physics, University of California, Santa Cruz, CA 95064, USA Received 29 March 1983
We study spin 5/2 fields on the lattice. Species doubling is found to be partially curable with an analogue of the KogutSusskind formalism, reducing the degeneracy from 16 to 4 while preserving local gauge invariance.
In a previous letter [1] we showed that spin 3/2 fields could be put on the lattice with an analogue of the Kogut-Susskind "staggered" fermion formalism [2]. The advantage of this formalism over an analogue of Wilson's method [3] is that local gauge invariance ("supersymmetry") is preserved. However, a spin 3/2 "staggered" fermion field describes 4 particles instead of only 1 (on a four-dimensional euclidean lattice). In this letter we generalize the construction o f "staggered" fermions to spin 5/2 fields. The continuum action for free, massless spin 5/2 fermions is given by [41
The naive lattice version of (1) is obtained by replacing the derivative a~ with the double lattice derivative (using standard notation)
a.f(x) -~ } [ f ( x + p ) - f ( x - p)] -~ Oufin the following.
(5)
The corresponding "naive" lattice action has the same species doubling problem as the naive lattice spin 1/2 [5] and spin 3/2 [l] actions, since the double lattice derivative (5) corresponds to sin pU in momentum space and the sine has 16 low momentum regions, so it would correspond to 16 particles. Starting from eq. (1) (where 3 is now the lattice derivative) we can diagonalize the v-matrices by writing [6]
O) The field ~uv is symmetric in the tensor indices and carries a Dirac index. We shall work in euclidean space and take the 7-matrices to be hermitian. We choose a representation where the ,),i are real and 7 0 is purely imaginary. In this representation,
..7
•
(2)
The action (1) is invariant under the following local gauge transformations
~..(x)
= r(x
+ ¢ + o)(7o) z~'xx × . . ( x ) ,
(6)
where
r(x) = (vo) x o (71)x~ (72) x2 (73)x3.
(7)
The matrices T(x) are unitary and satisfy
T+(x)'I'uT(x + fi) = flu(x),
(8)
with
6 t~uv = a~zXv + avXu ,
(3)
~0(x) = 1,
where ~ is a spin 3/2 field that satisfies 7"7t• = 0 ,
(4)
Supported by a grant from the National Science Foundation.
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
~l(X) = ( - 1 ) ~° ,
772(x) = ( - 1 ) xo+xl ,
r/3(x ) = ( - 1 ) x°+xl+x2 .
(9)
Substituting eq. (6) in the action (1) leads to 87
Volume 126B, number 1,2
S
~
T
PHYSICS LETTERS
The constraint (12) follows directly from eqs. (4) and (13). The action (10) has a continuous U(1) invariance,
,
x,indices
Xuv(x) -* exp[iae(x)]
-- ,Tv(X + ~)%(X + ~)'7~,(X + ~ + 5)GX~,~(X )
e(x) = ( - 1 ) x°+xl +x2 +x3 . •
(lO)
The field Xuv still carries a Dirac index a = 1, ..., 4, but since the action ( I 0 ) is diagonal in Dirac space we can retain only one of the components and by doing so we reduce the doubling from 16 to 4. The local supersymmetry (3), (4) for the field Xuv becomes 6Xuv(x) =
Ou#v(x) + Ovpu(x),
(11)
with
~u(x)pu(x)
=
O.
(12)
The field pu(x) is a "staggered" spin 3]2 field, defined in terms o f the field Xu(x ) as,
Xu (x) = r(x +/~) (3'0) x xx x+ 1 Pu ( x ) .
88
Xuv(x),
(14)
where
+ 2rtv(x +/])3xXxu(x)] + x~a [ ~ % ( x ) G x u u ( x ) - % ( x ) G x u d x ) l
23 June 1983
(13)
(15)
In the continuum limit this U(1) symmetry becomes embedded in the group 0(4). We foresee no problems in extending the above analyses to fields o f arbitrary non-integer spin. This is left as an exercise to the reader.
References [1] G. Maturana and C,P. van den Doel, Phys. Lett. 123B (1983) 332. [2] L. Susskind, Phys. Rev. D16 (1977) 3031. [3] K.G. Wilson, Phys. Rev. D10 (1974) 2445; in: New phenomena in subnuclear physics, Erice Lectures 1975. ed. A. Zichichi (Plenum, New York, 1977). [4] J. Fang and C. Fronsdal, Phys. Rev. D18 (1978) 3630. [5] L.H. Karsten and J. Smit, Nucl. Phys. B183 (1981) 103. [6] N. Kawamoto and J. Smit, Nucl. Phys. B192 (1981) 100.