- Email: [email protected]
Does noncommutative geometry encompass lattice gauge theory?
20 August 1998
Physics Letters B 434 Ž1998. 80–82
Does noncommutative geometry encompass lattice gauge theory? Meinulf Gockeler ¨
a,1
, Thomas Schucker ¨
b,2
a
b
Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Regensburg, D–93040 Regensburg, Germany Centre de Physique Theorique, CNRS - Luminy, Case 907, F–13288 Marseille Cedex 9, France ´ Received 15 May 1998 Editor: L. Alvarez-Gaume´
Abstract We are unable to formulate lattice gauge theories in the framework of Connes’ spectral triples. q 1998 Elsevier Science B.V. All rights reserved. PACS: 11.15 Keywords: MSC-91: 81T13 Yang-Mills and other gauge theories
1. Introduction Our fascination for Connes’ noncommutative geometry w1x has two sources: 1. It is general enough to treat continuous and discrete spaces on equal footing. 2. It has enough structure to include the Yang–Mills action. A natural question then is whether noncommutative geometry is compatible with standard formulations of lattice field theories w2x. In a more general frame of noncommutative geometry starting from differential algebras Dimakis, Muller–Hoissen ¨ & Striker w3x gave an affirmative answer. Since then Connes w4x completed the axiomatic foundation of noncommutative ŽRiemannian. geometry in terms of spectral triples. The triple consists of an associative 1 A ls o a t U n iv e rs ite´ d e P ro v e n c e . [email protected]. 2 A ls o a t U n iv e rs ite´ d e P ro v e n c e . [email protected].
E -m a il: E -m a il:
involution algebra A, a faithful representation r on a Hilbert space H and a selfadjoint operator D on H , ‘the Dirac operator’. In even dimensions one also requires the existence of a ‘chirality’, a unitary operator x on H of square one, that anticommutes with D. A real spectral triple has furthermore a ‘real structure’, an antiunitary operator J on H of square plus or minus one. These five items are to satisfy axioms. These axioms generalize properties of the commutative spectral triple of Riemannian spin manifolds M. There A s C `Ž M . is the commutative algebra of functions on spacetime, H is the space of square integrable spinors, D s Eu, x s g 5 and the real structure is given by charge conjugation. These axioms are tailored such that there is a one-to-one correspondence between commutative spectral triples and Riemannian spin manifolds. The items of the spectral triple allow to construct a differential algebra which is no longer chosen by hand. For the commutative spectral triple of a spacetime M this
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 7 3 4 - 5
M. Gockeler, T. Schuckerr Physics Letters B 434 (1998) 80–82 ¨ ¨
differential algebra is isomorphic to de Rham’s algebra of differential forms.
81
any kind of difference operator on the lattice as for instance the Dirac–Kahler operator, ¨ 4
DŽ x, y. s
Ý
i 2
hm Ž x . d y , xq mˆ y d y , xy mˆ .
Ž 6.
ms1
2. Any spectral triple for a lattice? Here We tried to construct a lattice action in terms of spectral triples starting from a lattice version of the Dirac operator. We failed already at the level of the axioms. More precisely, it is the first order axiom, D , r Ž a . , J r Ž a˜ . Jy1 s 0,
for all a,a˜ g A ,
Ž 1. that puts us out of business. On a smooth spacetime M this equation just says that the Dirac operator Eu is a first order differential operator. Consider a finite hypercubic lattice of N 4 points labeled by discrete 4-vectors x or y. We take for the algebra 4
A s [1N C 2 f ,
Ž 2.
my1
Ýx
n
hm Ž x . :s Ž y1 . ns1 ,
Ž 7.
and m ˆ denotes the lattice unit vector in m-direction. We define the chirality by the matrix
x Ž x , y . :s e Ž x . d x y ,
e Ž x . :s Ž y1 .
x 1 qx 2 qx 3 qx 4
Ž 8. Then, besides the first order axiom, all other axioms by Connes are satisfied, in particular Poincare´ duality. Indeed, if p x 4x g N 4 is the set of minimal projectors in A defined by p x Ž y . s d x y , then the intersection form, l x y :s tr x r Ž p x . J r Ž p y . Jy1 s e Ž x . d x y ,
Ž 9.
represented on 4
H s [1N C k 2 c ,
Ž 3.
by
Ž r Ž f . c . Ž x . :s f Ž x . c Ž x . .
.
Ž 4.
is non-degenerate. On the other hand Poincare´ duality fails for the naive lattice Dirac operator acting on k s 4 component spinors c Ž x ., 4
For later purposes we include k additional degrees of freedom that will be spin or colour. As a matrix the representation is r Ž f .Ž x, y . s f Ž x . d x y m 1 k . We take the real structure J to be such that J r Ž f . Jy1 s r Ž f . with z ¨ z meaning complex conjugation. ŽIn the concrete examples discussed below this will be fulfilled.. For the Dirac operator, we keep a general matrix D Ž x, y ., where the additional indices running from 1 to k are suppressed. Then the double commutator in the first order axiom becomes, D, r Ž f . , r Ž g . Ž x , y . s Ž f Ž y. yf Ž x . . Ž g Ž y. yg Ž x . . DŽ x, y. .
D Ž x , y . s 2i
The double commutator vanishes if and only if the Dirac operator is diagonal in Ž x, y .. This excludes
d y , xq mˆ y d y , xy mˆ g m .
Ž 10 .
ms1
Here the g m are the four Euclidean, Hermitian Dirac matrices. The chirality is x s 1 N 4 m g 5 and the real structure J is charge conjugation on each lattice point. Then the intersection form vanishes identically because tr g 5 s 0. At this point we recall that Connes’ geometric formulation of the standard model of electro-weak and strong interaction grew out of the two point space with Dirac operator Ds
Ž 5.
Ý
ž
0 m
m , 0
/
Ž 11 .
with m being the inverse distance between the two points. The two point space is a one dimensional lattice and this D is a kind of lattice Dirac operator.
M. Gockeler, T. Schuckerr Physics Letters B 434 (1998) 80–82 ¨ ¨
82
Still the standard model satisfies the first order condition and does so by adding antiparticles and strong interactions. We try to copy this trick: A s Ž C [ C . [ Mk Ž C . 2 Ž a,b,c . ,
Ž 12 .
H s Ž C [ C. m C k [ Ž C [ C. m C k ,
Ž 13 .
r Ž a,b,c . s
ž
Ds
xs
Js
ž
ž
0 m
ž
a 0
0 m 1k b 0
/
m m 1k 0
/
ž
0 m
0 m 1k 1
/
0
0 12 m 1k
12 m 1k
12 m c
0
,
0
0 -1 0
0
m m 1k 0
/
0 0 m 1k 1
Ž 14 .
0 0
,
Ž 15 .
,
Ž 16 .
ž
y1 0
/
( complex conjugation.
/
Ž 17 . Now the first order axiom works because the ‘colour’ Mk ŽC. is vectorlike. However the Poincare´ du-
ality fails, the intersection form is degenerate. Looking at the standard model, this is not that astonishing. Our model is necessarily closer to the standard model with right-handed neutrinos for which Poincare´ duality fails as well.
3. Conclusion Our conclusion is short and disappointing. We were looking for an overlap of lattice field theory and noncommutative geometry in the hope it would shed light on the definition of the functional integral within Connes’ geometry. But so far this overlap seems to be empty in Connes’ strict sense.
References w1x A. Connes, Noncommutative Geometry, Academic Press, 1994. w2x See, e.g., I. Montvay, G. Munster, Quantum Fields on a ¨ Lattice, Cambridge University Press, 1994. w3x A. Dimakis, F. Muller–Hoissen, T. Striker, Phys. Lett. B 300 ¨ Ž1993. 141; A. Dimakis, F. Muller–Hoissen, T. Striker, J. ¨ Phys. A 26 Ž1993. 1927. w4x A. Connes, J. Math. Phys. 36 Ž1995. 6194; A. Connes, Comm. Math. Phys. 155 Ž1996. 109, hep-thr9603053.
Physics Letters B 434 Ž1998. 80–82
Does noncommutative geometry encompass lattice gauge theory? Meinulf Gockeler ¨
a,1
, Thomas Schucker ¨
b,2
a
b
Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Regensburg, D–93040 Regensburg, Germany Centre de Physique Theorique, CNRS - Luminy, Case 907, F–13288 Marseille Cedex 9, France ´ Received 15 May 1998 Editor: L. Alvarez-Gaume´
Abstract We are unable to formulate lattice gauge theories in the framework of Connes’ spectral triples. q 1998 Elsevier Science B.V. All rights reserved. PACS: 11.15 Keywords: MSC-91: 81T13 Yang-Mills and other gauge theories
1. Introduction Our fascination for Connes’ noncommutative geometry w1x has two sources: 1. It is general enough to treat continuous and discrete spaces on equal footing. 2. It has enough structure to include the Yang–Mills action. A natural question then is whether noncommutative geometry is compatible with standard formulations of lattice field theories w2x. In a more general frame of noncommutative geometry starting from differential algebras Dimakis, Muller–Hoissen ¨ & Striker w3x gave an affirmative answer. Since then Connes w4x completed the axiomatic foundation of noncommutative ŽRiemannian. geometry in terms of spectral triples. The triple consists of an associative 1 A ls o a t U n iv e rs ite´ d e P ro v e n c e . [email protected]. 2 A ls o a t U n iv e rs ite´ d e P ro v e n c e . [email protected].
E -m a il: E -m a il:
involution algebra A, a faithful representation r on a Hilbert space H and a selfadjoint operator D on H , ‘the Dirac operator’. In even dimensions one also requires the existence of a ‘chirality’, a unitary operator x on H of square one, that anticommutes with D. A real spectral triple has furthermore a ‘real structure’, an antiunitary operator J on H of square plus or minus one. These five items are to satisfy axioms. These axioms generalize properties of the commutative spectral triple of Riemannian spin manifolds M. There A s C `Ž M . is the commutative algebra of functions on spacetime, H is the space of square integrable spinors, D s Eu, x s g 5 and the real structure is given by charge conjugation. These axioms are tailored such that there is a one-to-one correspondence between commutative spectral triples and Riemannian spin manifolds. The items of the spectral triple allow to construct a differential algebra which is no longer chosen by hand. For the commutative spectral triple of a spacetime M this
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 7 3 4 - 5
M. Gockeler, T. Schuckerr Physics Letters B 434 (1998) 80–82 ¨ ¨
differential algebra is isomorphic to de Rham’s algebra of differential forms.
81
any kind of difference operator on the lattice as for instance the Dirac–Kahler operator, ¨ 4
DŽ x, y. s
Ý
i 2
hm Ž x . d y , xq mˆ y d y , xy mˆ .
Ž 6.
ms1
2. Any spectral triple for a lattice? Here We tried to construct a lattice action in terms of spectral triples starting from a lattice version of the Dirac operator. We failed already at the level of the axioms. More precisely, it is the first order axiom, D , r Ž a . , J r Ž a˜ . Jy1 s 0,
for all a,a˜ g A ,
Ž 1. that puts us out of business. On a smooth spacetime M this equation just says that the Dirac operator Eu is a first order differential operator. Consider a finite hypercubic lattice of N 4 points labeled by discrete 4-vectors x or y. We take for the algebra 4
A s [1N C 2 f ,
Ž 2.
my1
Ýx
n
hm Ž x . :s Ž y1 . ns1 ,
Ž 7.
and m ˆ denotes the lattice unit vector in m-direction. We define the chirality by the matrix
x Ž x , y . :s e Ž x . d x y ,
e Ž x . :s Ž y1 .
x 1 qx 2 qx 3 qx 4
Ž 8. Then, besides the first order axiom, all other axioms by Connes are satisfied, in particular Poincare´ duality. Indeed, if p x 4x g N 4 is the set of minimal projectors in A defined by p x Ž y . s d x y , then the intersection form, l x y :s tr x r Ž p x . J r Ž p y . Jy1 s e Ž x . d x y ,
Ž 9.
represented on 4
H s [1N C k 2 c ,
Ž 3.
by
Ž r Ž f . c . Ž x . :s f Ž x . c Ž x . .
.
Ž 4.
is non-degenerate. On the other hand Poincare´ duality fails for the naive lattice Dirac operator acting on k s 4 component spinors c Ž x ., 4
For later purposes we include k additional degrees of freedom that will be spin or colour. As a matrix the representation is r Ž f .Ž x, y . s f Ž x . d x y m 1 k . We take the real structure J to be such that J r Ž f . Jy1 s r Ž f . with z ¨ z meaning complex conjugation. ŽIn the concrete examples discussed below this will be fulfilled.. For the Dirac operator, we keep a general matrix D Ž x, y ., where the additional indices running from 1 to k are suppressed. Then the double commutator in the first order axiom becomes, D, r Ž f . , r Ž g . Ž x , y . s Ž f Ž y. yf Ž x . . Ž g Ž y. yg Ž x . . DŽ x, y. .
D Ž x , y . s 2i
The double commutator vanishes if and only if the Dirac operator is diagonal in Ž x, y .. This excludes
d y , xq mˆ y d y , xy mˆ g m .
Ž 10 .
ms1
Here the g m are the four Euclidean, Hermitian Dirac matrices. The chirality is x s 1 N 4 m g 5 and the real structure J is charge conjugation on each lattice point. Then the intersection form vanishes identically because tr g 5 s 0. At this point we recall that Connes’ geometric formulation of the standard model of electro-weak and strong interaction grew out of the two point space with Dirac operator Ds
Ž 5.
Ý
ž
0 m
m , 0
/
Ž 11 .
with m being the inverse distance between the two points. The two point space is a one dimensional lattice and this D is a kind of lattice Dirac operator.
M. Gockeler, T. Schuckerr Physics Letters B 434 (1998) 80–82 ¨ ¨
82
Still the standard model satisfies the first order condition and does so by adding antiparticles and strong interactions. We try to copy this trick: A s Ž C [ C . [ Mk Ž C . 2 Ž a,b,c . ,
Ž 12 .
H s Ž C [ C. m C k [ Ž C [ C. m C k ,
Ž 13 .
r Ž a,b,c . s
ž
Ds
xs
Js
ž
ž
0 m
ž
a 0
0 m 1k b 0
/
m m 1k 0
/
ž
0 m
0 m 1k 1
/
0
0 12 m 1k
12 m 1k
12 m c
0
,
0
0 -1 0
0
m m 1k 0
/
0 0 m 1k 1
Ž 14 .
0 0
,
Ž 15 .
,
Ž 16 .
ž
y1 0
/
( complex conjugation.
/
Ž 17 . Now the first order axiom works because the ‘colour’ Mk ŽC. is vectorlike. However the Poincare´ du-
ality fails, the intersection form is degenerate. Looking at the standard model, this is not that astonishing. Our model is necessarily closer to the standard model with right-handed neutrinos for which Poincare´ duality fails as well.
3. Conclusion Our conclusion is short and disappointing. We were looking for an overlap of lattice field theory and noncommutative geometry in the hope it would shed light on the definition of the functional integral within Connes’ geometry. But so far this overlap seems to be empty in Connes’ strict sense.
References w1x A. Connes, Noncommutative Geometry, Academic Press, 1994. w2x See, e.g., I. Montvay, G. Munster, Quantum Fields on a ¨ Lattice, Cambridge University Press, 1994. w3x A. Dimakis, F. Muller–Hoissen, T. Striker, Phys. Lett. B 300 ¨ Ž1993. 141; A. Dimakis, F. Muller–Hoissen, T. Striker, J. ¨ Phys. A 26 Ž1993. 1927. w4x A. Connes, J. Math. Phys. 36 Ž1995. 6194; A. Connes, Comm. Math. Phys. 155 Ž1996. 109, hep-thr9603053.